The first scene in the movie begins with the song “Easy to Be Hard,” from the musical “Hair,” floating out of a car radio as two lovers meet. The song is a plea for people who espouse high principles of love and kindness to live up to those values in their personal lives. The sequence ends with gunshots and Donovan’s “Hurdy Gurdy Man,” a track that explicitly positions “songs of love” as a kind of babbling shtick. The dream is, quite literally, dead.

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The movie, and the search for the killer that follow, capture the limits of both law enforcement agencies and newspapers to find the truth.

The police departments in multiple jurisdictions unravel the Zodiac’s codes and develop leads, but they never amass sufficiently convincing evidence to make an arrest. At one point in the movie, they even struggle to convince a judge that they should get a search warrant. The reporters run through theories and chase down sources from multiple jurisdictions, but they don’t get any further. “Zodiac” is somewhat unusual, in that it’s a story where the police and reporters work together instead of at cross-purposes, but their team-up doesn’t take them further or produce any breakthrough that would have been impossible for them to reach if they were carrying out independent investigations.

These failures aren’t the result of hollowed-out institutions, though “Zodiac” does take place during the years when the rates at which serious crimes were being closed by the police declined. The characters have resources to pursue their investigations, and they’re given time and plenty of leeway by their superiors (though one local politician runs for governor on the argument that the cops didn’t have what they needed to crack the case). And neither is “Zodiac” a story about the sorts of failures involved in the Vietnam War, where brilliant people, restricted both by their own faith in technocratic solutions and their fears of being seen as soft on Communism, made fatally terrible decisions.

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Instead, Fincher captures the uncertainty and loss of confidence that follow from a prolonged failure by institutions and people who are doing everything they’re supposed to, only to find that it doesn’t produce the correct results.

“Zodiac” conveys this so convincingly because Fincher got three astonishing performances out of his lead actors.

The year before “Iron Man” returned him to Hollywood’s A-list, Robert Downey Jr. was first charmingly — and then worryingly — dissolute as San Francisco Chronicle reporter Paul Avery. When “Zodiac” begins, Avery is an experienced journalist with a bit of counter-culture edge, cool enough to school young editorial cartoonist Robert Graysmith (Jake Gyllenhaal), but still eager enough to get slammed on Aqua Velvas with his younger colleague.

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Avery’s spine melts into relaxation during his first visit to the bar with Graysmith, and he never quite gets it back. After he receives a Zodiac letter threatening him personally, Avery’s long hair, increasingly gray, stops looking like a deliberate fashion choice, and begin to look like the result of a man who can’t quite keep himself together. He sports one of the “I Am Not Paul Avery” buttons that became popular during the height of the Zodiac killer’s rampage, and takes wobbly aim at a shooting range. By the end of the movie, Avery’s living in a nest-like mess on a houseboat, wearing dirty clothes, taking hits of oxygen and ranting bitterly at Graysmith. Downey Jr. brought out the rage and vulnerability of Avery’s deterioration; he mocks Graysmith for his obsession, but Avery has shrunk his own life down to a tiny, defensible space.

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As Graysmith, Gyllenhaal’s large eyes — always one of his most powerful acting tools — make him look first like a woodland creature and later like a skull. When the movie begins, Graysmith is a winningly attentive single dad, cutely awkward in editorial meetings. But when he begins decoding his first Zodiac letter, using his fondness for puzzles to turn the sinister array of symbols into a decipherable but even more sinister message, it’s as if Graysmith can’t stop seeing codes and clues everywhere. (The song that’s playing over his commute to work is Santana’s “Soul Sacrifice.”) For a time, he can focus on Melanie (Chloë Sevigny), a fellow game enthusiast who becomes his second wife. It doesn’t last: Graysmith becomes edgy, nocturnal, as if he shuts his eyes even for a moment he’ll miss something vitally important, the piece of information that will allow him to understand everything.

And San Francisco detective Dave Toschi (Mark Ruffalo) begins to wilt under the pressure of his own image. Even as he becomes the inspiration for Harry Callahan (Clint Eastwood) in the “Dirty Harry” movies, he’s unable to find definitive evidence that would link his best suspect, Arthur Leigh Allen (John Carroll Lynch) to the killings, and when Toschi meets the man in person, Allen is a polite, unflappable blank. Toschi starts to look hunched, as if the iconic shoulder holster that once seemed to pull his shoulders back and throw out his chest, and that Steve McQueen imitated in “Bullitt,” is starting to weigh him down.

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When Graysmith finds his way to Allen at the end of the movie, the confrontation is decidedly anti-climatic. “Can I help you?” Arthur, now working as a clerk, asks him. “No,” Graysmith tells him. To a certain extent, that’s because the only thing that matters is Graysmith’s own sense of certainty. It’s also true, though, that Allen is never going to confess; he’s never going to provide either Graysmith or the audience with the definitive ending we crave so badly.

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Ten years after “Zodiac” was released, and almost fifty years after the July 4 killing that sets the movie in motion, we’re still living with and working through the consequences of the decline and loss of faith David Fincher captured in this masterful film. Fincher’s “Fight Club” offered up a vision of weaponized male turned against society as a whole, while his “Gone Girl” portrayed female anger that had been distilled into a particularly venomous domestic poison. “Zodiac” is his movie for the rest of us, who have to live in a world going slowly insane without losing ourselves.