L-functions are at the heart of the LMFDB. What are they? We will give a brief survey, referring to number theory textbooks for details.

The simplest L-function is the Riemann zeta function \(\zeta (s)\). This

is a complex analytic function (apart from a pole at \(s=1\));

has a Dirichlet series expansion over positive integers (valid when \(\mathfrak {R}(s)>1\)): $$\begin{aligned} \zeta (s)=\sum _{n=1}^{\infty } \frac{1}{n^s}; \end{aligned}$$

has an Euler product expansion over primes p (when \(\mathfrak {R}(s)>1\)): $$\begin{aligned} \zeta (s)=\prod _p\left( 1-p^{-s}\right) ^{-1}; \end{aligned}$$

satisfies a functional equation : $$\begin{aligned} \xi (s)=\pi ^{-s/2}\varGamma (s/2)\zeta (s) = \xi (1-s); \end{aligned}$$

has links to the distribution of primes.

L-Functions: A Definition

The definition of an L-function encapsulates these properties: it is a complex function with a Dirichlet series and an Euler product expansion which satisfies a functional equation. There are other more technical axioms (by Selberg) which we omit here: refer to the LMFDB’s own knowledge database for details: http://www.lmfdb.org/knowledge/show/lfunction.

Some of the defining properties have not in fact been proved for all the types of L-function in the database: this can be very hard! For example, Andrew Wiles proved Fermat’s Last Theorem by proving the modularity of certain elliptic curves over \({\mathbb {Q}}\), which amounted to showing that the L-functions associated to elliptic curves really are L-functions in the above sense. This is not yet known in general for elliptic curves defined over other algebraic number fields.

Other expected properties of L-functions are not even known for \(\zeta (s)\). For example, the Riemann hypothesis concerning the zeros of \(\zeta (s)\) has remained open since it was formulated by Riemann in 1859.

The Riemann Hypothesis

The Riemann Hypothesis states that all the “non-trivial” zeros of \(\zeta (s)\) (excluding those coming trivially from poles of \(\varGamma (s)\)) are on the “critical line” \(\mathfrak {R}(s)=1/2\).

This was (part of) Hilbert’s 8th problem and is also one of the Clay Mathematics Institute Millennium Prize Problems, so a million dollars awaits the person who proves it. There are similar conjectures about the location of the zeros of all L-functions, which are collectively known as the Generalized Riemann Hypothesis (GRH). These are not only of theoretical (or financial!) interest, but have important applications to the complexity of computing important quantities in number theory. For example, computing the class number of a number field is much faster if one assumes GRH for the number field’s own L-function, its Dedekind \(\zeta \) -function.

What can a database say in relation to this problem?

It can give the object its own web page (http://www.lmfdb.org/L/Riemann/) which shows basic facts about it, and its graph along the critical line \(1/2+it\) to “show” the first few zeroes. This is a pedagogical function of the database.

It can also store all the zeroes which have so far been explicitly computed: there are more than \(10^{11}\) (that is one hundred billion) of them at http://www.lmfdb.org/zeros/zeta/, all computed to 100-bit precision by David Platt (Bristol), who in 2014 won a prize for his contributions to progress on the Goldbach Conjecture. This resource can then be used to study properties of the zeroes, such as their distribution, and connections to random matrices, showing that the database also serves as a research tool.

Degrees of L-Functions

The Euler product for a general L-function has the form

$$\begin{aligned} L(s) = \prod _p 1/P_p\left( 1/p^{s}\right) \end{aligned}$$

where each \(P_p(t)\) is a polynomial, and the product is over all primes p. These polynomials all have the same degree, called the degree of the L-function, except for a finite number indexed by primes dividing an integer called the conductor of the L-function, where the degree is smaller. The zeros of these polynomials are also restricted in a way depending on another parameter, the weight.

For example, \(\zeta (s)\) has \(P_p(t)=1-t\) for all primes p; the degree is \(d=1\), and the conductor is \(N=1\).

L-Functions of Degree 1

There are other L-functions of degree 1, with larger conductor N, which have been studied since the nineteenth century: Dirichlet L-functions. Their Dirichlet coefficients \(a_n\) are given by the values of a Dirichlet character \(a_n=\chi (n)\), meaning that they are multiplicative and periodic with period N.

An example with \(N=4\) is

$$\begin{aligned} L(\chi ,s)=1^{-s}-3^{-s}+5^{-s}-7^{-s}+-\cdots , \end{aligned}$$

with all even coefficients 0 and the odd coefficients alternating \(\pm 1\). Dirichlet used such L-functions to prove his celebrated theorem about primes in arithmetic progressions: for any integers \(N\ge 1\) and a, there are infinitely many primes \(p\equiv a\pmod {N}\), provided that a and N are coprime. The previous example can be used not only to show that there are infinitely many primes \(p\equiv 1\pmod 4\) (for which \(\chi (p)=+1\)) and infinitely many primes \(p\equiv 3\pmod 4\) (for which \(\chi (p)=-1\)) , but also to show that (in a precise sense) the primes are equally distributed between these two classes.

This is a complete list of all L-functions of degree 1. For degrees greater than 1, a complete classification has not yet been established, though a wide variety of sources of L-functions is known, and in some cases (such as in degree 2, see below), we conjecture that all L-functions do arise from these known sources.

Other Sources of L-Functions

A wide variety of mathematical objects have L-functions: algebraic number fields, algebraic varieties (including curves). There is a general term motive for objects which have L-functions.

In many cases, while we know how to define the L-function of a more complicated object, it has not yet proved that it actually satisfies the defining axioms for L-functions. Even for elliptic curves over \({\mathbb {Q}}\), this would have been true until the mid-1990s; for elliptic curves over real quadratic fields such as \({\mathbb {Q}}(\sqrt{2})\) it was true until 2013! Now, these elliptic curves are known to be modular [5].

L-Functions of Number Fields

An algebraic number field, or simply number field, is a finite extension of the rational field \({\mathbb {Q}}\), such as \({\mathbb {Q}}(\sqrt{2})\) or \({\mathbb {Q}}(i)\) or \({\mathbb {Q}}(e^{2\pi i/m})\). Every number field K has an L-function called its Dedekind zeta function \(\zeta _K(s)\), defined in a similar way to Riemann’s \(\zeta (s)=\zeta _{{\mathbb {Q}}}(s)\), and with similar analytic properties.

Just as the analytic properties of \(\zeta (s)\) imply facts about the distribution of primes, from the analytic properties of \(\zeta _K(s)\) we can deduce statements about prime factorizations in the field K. For example, taking \(K={\mathbb {Q}}(e^{2\pi i/m})\) we can prove Dirichlet’s Theorem on primes in arithmetic progressions using a combination of algebraic and analytic properties of \(\zeta _K(s)\).

Also, just as some properties of \(\zeta (s)\) are not yet proved (e.g. the Riemann Hypothesis), the same is true for \(\zeta _K(s)\): the Generalized Riemann Hypothesis or GRH remains unsolved.

L-Functions of Curves

Algebraic curves defined over algebraic number fields also have L-functions, whose degree depends on both the degree of the field over which the curve is defined and the genus of the curve. So an elliptic curve over \({\mathbb {Q}}\), which is a curve of genus 1 defined over a field of degree 1, has a degree 2 L-function, elliptic curves over fields of degree d have L-functions of degree 2d, and so on.

It is widely believed that all degree 2 L-functions arise as follows: they either are products of two degree 1 L-functions, or come from elliptic curves over \({\mathbb {Q}}\), or from (a special kind of) modular form. The insight of Weil, Taniyama, Shimura, and others in the 1960s and 1970s was to realize that the latter two sources actually produce the same L-functions! This insight is behind the famous theorem of Wiles et al. that “every elliptic curve (over \({\mathbb {Q}}\)) is modular”, from which Fermat’s Last Theorem was a consequence. But it is still an unsolved problem to show that those degree 2 L-functions which are not products of degree 1 L-functions do all arise from automorphic forms.

Higher Degree L-Functions

For degrees 3 and 4, we do not yet even have a conjecture concerning all sources of L-functions, and for those which are known, not all the conjectured connections between them have been proved.

We mentioned above the recent result [5] that elliptic curves defined over real quadratic fields (such as \({\mathbb {Q}}(\sqrt{5})\)) are modular. This means that two sources of L-functions of degree 4: on the one hand, elliptic curves over such a field, and on the other hand Hilbert modular forms over the same field, actually produce the same L-functions. Such results are extremely deep and require a vast amount of theory to establish, including real, complex, and p-adic analysis and algebra, as well as some explicit computations (the ArXiV version of Freitas et al. [5] includes a number of Magma scripts).

By contrast, over imaginary quadratic fields (e.g. \({\mathbb {Q}}(\sqrt{-1})\)) we conjecture, but cannot prove in general, that elliptic curves have L-functions also attached to a different kind of modular form, Bianchi modular forms. These can be computed, and work is in progress in entering many examples into the LMFDB, even though they are not all known to “be modular” and hence have genuine L-functions.

Modularity of individual elliptic curves over imaginary quadratic fields can be proved using the Serre–Faltings–Livné method (which uses Galois representations rather than analysis) as explained in a 2008 paper [6] by Dieulefait, Guerberoff, and Pacetti. We are currently using their method to prove modularity of all the curves in the database; at the same time we are developing enhancements to the algorithm to make it more efficient. A theoretical proof that all elliptic curves over these fields are modular seems very far off, so even in the world of L-functions of degree 4 it is still important to carry out experiments and collect data.

Showing Connections Through the LMFDB

The LMFDB shows connections between different objects with the same L-function, such as those described above, by linking its databases of (for example) elliptic curves over real quadratic fields, and Hilbert modular forms over the same field. The home page of each elliptic curve includes a link to the associated Hilbert modular form, and to the associated L-function, and (in progress) vice versa.

One difficulty we have encountered in setting up these links on the website, which is perhaps typical in a large project where many different individuals are providing data, is to maintain consistency of labelling of objects. Over the field \({\mathbb {Q}}(\sqrt{5})\), the Hilbert modular forms were computed (in Magma) by John Voight (Dartmouth College) and Steve Donnelly (Sydney) [7], while the elliptic curves were computed (in SageMath) by Jonathan Bober (Bristol), William Stein (Washington), Alyson Deines (CCR), and others [8]. These groups used essentially the same naming convention, but we were careful to check that the labels of matching objects did match exactly, resulting in one set of data (the elliptic curves) requiring relabelling.