For her new series and character, Michelle Dockery — Lady Mary of “Downton Abbey” fame, the most superior and (let’s be honest) tiresome of the Crawley girls — has crossed the ocean, traveled forward in time (to now), become American and acquired a prison record. She’s kept some of the haughtiness that seems in some way her birthright — and the cheekbones and eyes and all that — but within a character that could otherwise not be more different from the one that dominated her professional life for six seasons.

In the smartly played, agonizingly suspenseful “Good Behavior,” which premieres Tuesday on TNT, Dockery plays Letty Raines, a con woman and thief newly on parole, but already back conning and thieving, and backsliding as well, in her attempts to stay clear of drugs and alcohol and reconnect with her young son. Terry Kinney plays her parole officer, a not untroubled soul going out of his way, and risking his career, to help her stay out of jail.

That is not, however, the main order of business. While robbing a hotel room, Letty finds herself forced to hide in a closet and overhears a conversation between a contract killer (Juan Diego Botto as Javier Pereira) and his client. Attempting to head him off at the pass, she winds up both his prisoner and his accomplice. (Details too spoilery to spill.)

In spite of some humdrum fanciness — the fast-slow alterations in camera speed, the natural desire of designers and photographers to make everything pretty — the series, which is set (so far) in South Carolina and Tennessee, is first-class Southern noir; thick with atmosphere and weather and spreading the suspense around so that most every character we meet seems either in danger or a danger. But the action stays life-sized and plausible; the talk is largely crisp, rarely overripe.

Most of what keeps Letty in Javier’s company, in the three episodes available for review, is of the “I will find you and kill you if you leave” variety. (“Yeah, I get it. You know how to train a dog,” she says when he tells her this once again.) But the characters, as characters will, also get something from each other. His life is ordered and disciplined, where hers is chaos, a constant improvisation she tries to balance with a motivational app that offers thoughts such as “I have a lot to be proud of today” and “I am the best me I can be.” (“Is this a friend of yours?” asks Javier, who is from Argentina, where self-help is possibly not so prevalent. “This woman who talks to you.”)

That is also the premise of “The Odd Couple.”

There is some basic-cable sex in it, but nothing in the early stages to suggest romance, beyond the show business physics of their mutual, comparable hotness. As the story goes on, and things go inevitably wrong, a little humor creeps in here and there; there is relief in small, beside-the-point moments.

He’s a killer, she’s a crook, and by the old laws of pop-cultural justice, both should get their comeuppance. Both might, of course, but that this is a series — based on a series of novels by Blake Crouch, adapted for television by show runner Chad Hodge, who also adapted Crouch’s “Wayward Pines” trilogy for Fox — suggests it will not be soon in coming.

At the same time, we have become accustomed to bad behavior as a dramatic engine for television shows in the nearly 20 years, if you can believe that, since “The Sopranos” debuted. And before that there were the novels of Jim Thompson and James M. Cain and Patricia Highsmith and films of the French New Wave upending and inverting the old heroic conventions.

Do we mind that they’re bad? Are they bad? What do we mean by “bad”? These are not questions that “Good Behavior” tentatively asks on our behalf.

“That’s it,” says Letty to Javier, “It’s your job, to kill people. You don’t care.”

“I care,” says Javier. But we are left to wonder what he means.