But they still had to render a separate verdict on whether to impose it.

Defense experts testified that they had diagnosed the gunman, Robert Bowers, 50, with schizophrenia and other serious mental disorders, saying that he had signs of “permanent brain damage” and that he suffered from paranoia and delusions.

Experts called by the prosecution disputed the findings about schizophrenia and delusions, arguing that Mr. Bowers believed in racist ideas that are widespread.

His defense team includes Judy Clarke, a lawyer with a long record of defending people accused of capital crimes, including the Unabomber, one of the Boston Marathon bombers, and the man who opened fire in an Arizona grocery store parking lot, killing six people and injuring 13, including former Representative Gabrielle Giffords.

His lawyers repeatedly but unsuccessfully challenged the government’s intention to seek the death penalty. In a filing this year, the defense argued that under Attorney General Merrick B. Garland, the Justice Department had been arbitrary in deciding whether to pursue capital punishment. They cited hundreds of murder cases in which Mr. Garland had elected not to seek the death penalty, including the 2019 mass shooting by an anti-immigrant extremist in a Walmart in El Paso.

The government rebutted these arguments by insisting that there were factors in this case, such as Mr. Bowers’s open antisemitism and his decision to attack during a worship service, “that make the death penalty specifically warranted here.”

During the penalty phase of the trial, witnesses described how Mr. Bowers’s already fragile mental state started to spiral precipitously starting in 2014, when he lost his grandfather, his home and his one close friend in quick succession.

One forensic psychiatrist, who examined Mr. Bowers for nearly 40 hours and diagnosed him with schizophrenia, said he had become obsessed with notions about Satan and an apocalyptic race war, a delusion that propelled his decision to attack the Pittsburgh synagogue.