The cinematic stories of heroin-addicted musicians tend to fall into two categories. There’s the recovery narrative, in which someone like Ray Charles hits bottom, cleans up and enters a new phase of healthy creativity. Then there’s the tragedy narrative, which covers doomed icons like Charlie Parker, Sid Vicious and — well, that’s a long list.

Then there are the puzzling outliers who remained fitfully functioning heroin addicts their entire lives. That’s a much shorter list. One would be Art Pepper, the saxophonist who for decades alternated between prison and the recording studio. (His 1980 autobiography was cheekily titled “Straight Life.”) Another would be Chet Baker, the jazz trumpeter and vocalist played by Ethan Hawke in Robert Budreau’s fictionalized biopic “Born to Be Blue.”

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A romantic figure who ranks somewhere between cult icon and jazz giant, Baker helped spearhead the West Coast Cool sound (along with Pepper and others) that briefly overshadowed East Coast firebrands like Miles Davis (Kedar Brown, glowering) and Dizzy Gillespie (Kevin Hanchard, all smiles). Drugs derailed Baker’s career — somewhat indirectly — when in 1966 he lost his front teeth in a drug-related beating. “Born to Be Blue” picks up around that time, following Baker as he tries to get clean, learns to play through dentures and pumps gas to pay his bills.

“This is the first time I’ve ever seen him sweat,” says Dick Bock, a real-life record producer played with flinty intelligence by an excellent Callum Keith Rennie. Baker needs work to keep his parole officer at bay, but he also has a new love, Jane. Although she’s a fictional, composite character, she’s given a mind and a soul by Carmen Ejogo (“Selma”). She and Hawke make an endearing bohemian couple living cozily in a Volkswagen van near the Southern California beach bluffs.

Hawke is in fine form as the talented, sweet-natured, self-destructive Baker, and his vocal rendition of “My Funny Valentine,” Baker’s achingly sad signature song, is a highlight of this somewhat uneven film. “Born to Be Blue” is a bit like Baker himself: a little sloppy and ragged around the edges, but ultimately compelling.