Towards the end of the new limited series “The Investigation,” the show’s central police inquiry seems stalled. Nothing is moving in the right direction, and all players seem understandably frustrated. One cop (Laura Christensen) sets things right with an unlikely note of encouragement. “This isn’t a perfect crime,” she tells her boss (Søren Malling). “It’s a clumsy, disgusting crime. So we must have overlooked something.”

This is an unexpected note for the TV procedural — which thrives on the sleuth as genius, seeing through a case that comes as a shock to the viewer. But “The Investigation,” written and directed by Tobias Lindholm and airing stateside on HBO after having debuted overseas, is no ordinary show. For one thing, the outcome will be known already to readers of the news: These Danish-speaking cops, in Copenhagen, are looking into the real-life 2017 killing of Swedish journalist Kim Wall, reported missing after she boarded a submarine to interview its owner. For another, the work of the police, facing down a crime sloppily committed but difficult to pin down, is allowed to proceed with a grinding forward momentum. It’s a pace that can be punishing but that also makes gains that much more impressive. That “The Investigation” moves so methodically through its story is a sign of respect for subjects and viewers alike — it’s a show that understands the gravity of its own story. Its intense unity of vision makes “The Investigation” the first great scripted series of 2021.

A healthy share of “The Investigation’s” success is owed to its cast. Malling, here, plays Jens Møller, the head of homicide in the Copenhagen police department. The performance is one suffused with duty; as Møller’s personal life suffers and as his psyche is overtaken with the monthslong quest for justice, one senses him pushing himself forward. Pilou Asbæk of “Game of Thrones” delivers an only slightly flashier performance as prosecutor Jakob Buch-Jepsen, who acts as something of a story engine: He can only bring the case forward if Møller and team manage certainty on a case seemingly designed to frustrate. Those frustrations find human expression in the late Kim’s parents Ingrid and Joachim Wall (Pernilla August and Rolf Lassgård) whose grief is given a dignity, and — lacking a place to put it or a way to express it — a sort of oceanic shapelessness. In one strikingly shot sequence at the midpoint of the series, we see them facing the water that, somewhere, holds their daughter, collapsing in a sort of unsolvable anguish. As we move from close-up to a long shot with water lapping their feet, the parents are dwarfed by sea, and by the pain within.

Their hopelessness lends urgency to the quest, one that, early on, sees Møller writing four columns on a whiteboard: “natural death / accident / suicide / homicide.” (Lindholm, who never indulges showiness in his shots, knows his way around a sort of bleak rigor: in addition to his work for the big screen and as a writer of the Danish series “Borgen,” he directed episodes of “Mindhunter.”) In order to determine Wall’s death was occasioned by the final of these, his team must find, in the waters off the Danish coastline, the pieces of Wall’s body. This is the series at its grimmest, both for the manner in which the boundless ocean conspires against the case ever being solved and for certain specifics: Buch-Jepsen tells Møller, at one point, “I can’t dismiss the accused’s explanation if we don’t find the head and any lesions.”

Uplift is hard to come by here. But there’s a sense of wonder glimmering around the edges of the process: for instance, in searching for the head, Møller has dispatched some 100 volunteers to search the Danish and Swedish coasts. Christensen’s Maibritt Porse has a haunted forward stare that instructs you as to the meaning she finds in closing this case; the scenes she and Malling share are among the series’s best, infused with a sort of slow-burning anger that’s been metabolized into the quest for justice.

That impulse towards justice is a human one, and one that can be put towards troubling ends: So often, crime TV has its roots in the gleefully punitive. So many shows provide a giddily indulgent look at the dark side of humanity. Here, though, Wall’s killer is neither seen nor mentioned by name throughout the series. It may seem lofty to say that the cops’ primary objective is to solve the case rather than prosecute the killer — a prosecutor is involved, after all — but the show ends up convincing its viewer, beat by painstaking beat. Its subject matter is, tragically, perfectly apt: Wall was known in life for superlative work as a journalist, bringing underexposed cases to light through rigorous reporting. The similar rigor of those seeking to bring closure to the far-too-short story of her life was one way of honoring her. “The Investigation,” so deeply a tribute to what is good — self-sacrificing, humble, serious when it counts — about the communities we share, is another.

“The Investigation” premieres on HBO Feb. 1 at 10:00 p.m. E.T./P.T.

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