A new poll isn’t exactly a birthday present that was on Sen. Dianne Feinstein’s wish list.

Feinstein, who turned 85 on Friday, is coming off a smashing 44 to 12 percent primary win over fellow Democrat Kevin de León, a termed-out state senator from Los Angeles.

But the poll by the University of Southern California and the Los Angeles Times released this week shows that Feinstein, a former mayor of San Francisco, still hasn’t closed the deal with a lot of California voters.

The post-primary survey found the four-term senator with a 36 to 18 percent lead over de León in the November election, with a whopping 46 percent of registered voters undecided.

Those numbers come with some caveats.

First, just because someone is registered to vote doesn’t mean that person will actually cast a ballot in the fall. Turnout was only about 35 percent in the primary, although November elections bring out more voters. Feinstein’s numbers look better when only those who voted June 5 are considered, with her lead increasing to 49 to 26 percent over de León, and the undecided falling to 25 percent.

Second, 54 percent of Republicans are undecided, along with 55 percent of independents. With no Republican in the Senate race, plenty of die-hard GOP voters probably don’t want to decide which Democrat they like more and will just skip the race. And independents typically turn out in lower numbers than partisan voters, even in general elections.

Even so, Feinstein’s inability to crack the 50 percent mark in a head-to-head matchup with de León has to be worrisome for someone whom Californians have been electing to the Senate since 1992.

While there’s no indication Feinstein is in any real danger of being retired on Nov. 6, the poll provides at least a sliver of hope for de León, who can argue to supporters that there’s still a chance lightning can strike. Those many undecided voters could decide that de León’s progressive plans fit them better than Feinstein’s more moderate stances, although that’s a long-shot bet.

Still, what any incumbent wants going into a general election is an air of invincibility, and that’s not something the new poll gives Feinstein.

“More than four months before the election, polling reflects an electorate that knows Dianne Feinstein and does not know Kevin de León,” said Jonathan Underland, a spokesman for de León. “We see room to grow in a one-on-one race over the next five months.”

Feinstein’s people don’t see anything to worry about in the poll.

“She’s got a big lead and is ahead with every voter group,” said Bill Carrick, Feinstein’s longtime consultant. “The poll reflects the same sort of margin we’ve had in the past.”

John Wildermuth is a San Francisco Chronicle staff writer. Email: jwildermuth@sfchronicle.com Twitter: @jfwildermuth