So enduring is the mythology surrounding John Dillinger that even today, 75 years after the bank robber was gunned down by federal agents, there is talk about his extraordinary sexual prowess and, moreover, mystery about the eventual whereabouts of his penis.

Dillinger’s audacious string of robberies and prison escapes in the early 1930s turned him into an American folk hero, a Depression-era Robin Hood. He and his gang robbed more than a dozen banks between May 1933 and July 1934, stealing more than $300,000, the equivalent of more than $5m today. He also destroyed thousands of mortgage records during the robberies, helping many poor people escape onerous payments to rapacious banks.

Newspapers received thousands of letters from people devastated by the ravages of the Depression praising Dillinger as a man who “robbed those who became rich for robbing the poor”, a man who “isn’t half as cheap as a crooked banker or crooked politician”, sentiments that have surprising resonance today.

Reports of Dillinger’s sexual prowess stem, in part, from the reputation the charming, athletic thief — who became known as “Gentleman Johnny”, always well dressed, groomed and manicured — had for calmly flirting with female customers while he was in the middle of a robbery. “He was known for little bits of courtesy, especially towards women,” said Bryan Burrough, who wrote Public Enemies, the book on which the new film is based. “He would josh and joke during bank robberies. He had obvious charisma, obvious star power.”

As for the physical dimensions of his genitalia, a photograph taken in the morgue of Dillinger’s corpse, draped in a sheet, appeared to show such a stupendous penis that many newspapers felt compelled to black it out when they reprinted the photograph. The most likely explanation is that the offending member was, in fact, Dillinger’s arm under the sheet, stiffened in rigor mortis. The autopsy apparently showed no such genital exaggeration. Still, that and many other Dillinger legends persist. Some believe his penis has been secretly stored in Washington, at the Smithsonian, the country’s leading repository of Americana. “We have no penises,” the museum told The New York Times when inquiries were made a few years ago. It also seems not to have ended up at the museum in the Walter Reed vete­rans’ hospital, as some believe. “We are not the home of the Dillinger penis,” an official there said.

The most intriguing possibility is that its final resting place was a pickling jar in the office of the FBI director J Edgar Hoover.

The histories of Dillinger and Hoover are inextricably linked. It was Hoover who designated the suave bank robber America’s first Public Enemy No 1. The hunt for Dillinger and other gangsters of the early 1930s — Hoover’s “War on Crime” — and their eventual downfall were critical in helping Hoover to establish the FBI, previously an obscure arm of the Justice Department, as the immensely powerful nationwide law enforcement agency it still is today.

“The War on Crime was the birth of the modern FBI,” Burrough says. “It was a transformation from a bunch of obscure law-enforcement types without arrest powers. In the span of 20 months, they became the G-Men of lore, with real and perceived power.” It also helped Hoover to establish himself as the most feared figure in 20th-century America, whose knowledge of secrets in high places helped him to blackmail his adversaries and allowed him to maintain power. He was director of the FBI from 1935 until 1972.

Hoover is known to have kept a number of Dillinger mementos in a glass case at FBI headquarters, including his death mask, his Colt semi-automatic, his trademark straw hat, change from his pockets and his sunglasses. The possibility that Hoover might also have kept Dillinger’s penis as a totemic trophy seemed less surprising when it was widely reported after his death that the ruthless director of the FBI for 37 years had secretly been a cross-dressing homosexual. But it has never been found.

The son of a tough Indianapolis grocer, Dillinger was often in trouble as a kid and got involved in a neighbourhood gang called the Dirty Dozen. His life changed after he robbed a shop when he was 20 and served nearly nine years in prison, a brutal sentence for a first offence. It was there that he learnt about robbing banks from an inmate called Herman Lamm, who had been in the German army and believed military tactics could be successfully applied to bank robberies.

Dillinger learnt to case banks before a job, sometimes pretending to be a movie company looking to use the bank as a location. His gang, which refrained from alcohol before robberies, also planned escapes minutely, aided by maps, which had only recently been published. They hid petrol in haystacks along their escape routes.

Dillinger’s spectacular crime spree, which was avidly followed by the public in news­papers and movie newsreels, made Hoover and his men a nationwide laughing stock, particularly after a shoot-out in which they killed innocent bystanders, but let Dillinger and his gang escape. “We were a bunch of greenhorns who had no idea what we were doing,” one agent admitted. Hoover was furious when he heard that audiences were cheering a Warner Bros newsreel about the manhunt whenever Dillinger was shown, and booing and hissing at his agents.

Dillinger, however, had few illusions about his fate. “I’m travelling a one-way road,” he told a friend, “and I’m not fooling myself as to what the end will be. If I surrender, I know it means the electric chair. If I go on, it’s just a question of how much time I have left.”

Hoover was thrilled when Dillinger was killed on the night of July 22, 1934, outside the Biograph cinema, in Chicago, in an operation led by special agent Melvin Purvis. The gangster was only 31. According to legend, before Dillinger was shot, Purvis called out to him: “Stick ’em up, Johnny, we have you surrounded.”

Although a number of people died during the robberies, Dillinger is believed to have killed only one man himself. Despite being accused of murder, he had become such a popular figure that when he was killed, people crowded round to dip their handkerchiefs and skirts in his blood. Two days later, 5,000 people attended his funeral.

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