Unlike Paul, Jesus was not-as far as we know-a Roman citizen. On the Roman understanding of things, Jesus was technically the citizen of a client kingdom, and therefore subject to that kingdom's laws. In the gospels, Pilate does indeed attempt to send Jesus to his king (Herod), who then rejects the whole case and sends Jesus back to Pilate.

But really, what precise law Jesus had broken was probably not at the top of anyone's mind. If we follow the gospel accounts, there's plenty of good reason for the Sanhedrin and Pilate to be willing to kill Jesus (N.B. I am not claiming that the gospels are historically accurate. But issues of biblical exegesis lie beyond the scope of this subreddit, and for our purposes what matters is that even we take the very pro-Jesus gospel narratives at their word we can still understand why authorities had a problem with Jesus).

Jesus is coming into Jerusalem for Passover, a holiday that drew Jews from across the Mediterranean to Jerusalem. This was not a happy time for the Romans. The Jews had never bore the yoke of Roman rule especially well, and when their holiest city was packed with the most zealous members of their faith, riots had a tendency to break out. Indeed, Pilate is probably in Jerusalem specifically because the situation is so volatile (Romans typically hated the city and most Roman governors preferred to live in other, more Hellenic, cities in Judea).

So the city and her authorities are on edge. Then, according to the gospels, Jesus enters and is greeted by lines of people throwing down palm branches and cloaks. The Romans call this sort of ceremony an adventus, and it is carried out when important persons come to a city. In the Hellenistic period, it was done for kings. The Sanhedrin is understandably panicked. It looks as if the crowds are willing to crown Jesus king, in which case Pilate will order out his soldiers, and they will likely kill and pillage indiscriminately until they are reined back in. As the local elite, the Sanhedrin have the most to lose should this occur. So they arrest Jesus, accuse him of the crime they can (blasphemy) and send him to Pilate to execute (they had no right to carry out executions themselves).

Now comes Pilate's turn to panic. If he executes a man beloved by the crowd, then it's pretty likely the crowd will riot. Caesar will not be pleased. So he tries to pass the buck by claiming he has no legal jurisdiction. But Herod wisely refuses to take on the onus of ordering the execution, so back Jesus goes to Pilate. Pilate then arranges an act of clemency, offering the people a chance to free either Jesus or another man. Perhaps to Pilate's surprise, the crowd opts for the other man. Now slightly more confident, Pilate orders Jesus flogged (as governor he's really got carte blanche to do this, unless the accused is a Roman citizen). Once more, the crowd does not riot. So Pilate allows the execution, with one final ostentatious show of blamelessness ("I wash my hands of this") in case the crowds have second thoughts once they actually see Jesus on the cross.

So Jesus is executed. The placard above him reading "King of the Jews" is testimony to what the Sanhedrin and the Romans thought he was aiming at. Jesus dies quickly (crucifixion often takes days to kill). When one of Jesus' followers asks permission to take down the body, Pilate quickly assents. Normally bodies are left to hang on crosses, but the Jews have strict laws about such things and, in any event, its probably best that the body of the would-be king is taken down before it stirs people up.

From Pilate's perspective, the situation is resolved. The city survives Passover without a riot, and all it cost was the life of Jewish peasant, who was probably looking to stir up trouble anyway.

Drawing from 1 Raymond E. Brown, The Death of the Messiah 711-12, 717-19, 850-51 (1994):

Narratives from the New Testament regarding Jesus's trial must be approached cautiously, as the goal of the Gospels was to dramatize the religious—not legal—meaning of Jesus's condemnation. Because of their purpose, the Gospels relate few legal details of Jesus's trial by Pilate. Additionally, none of the Gospels even claims that sympathetic eyewitnesses were present at the trial. As such, any historical information the Gospels offer would be derived from hearsay, post factum explanations from authorities, or educated guesses. Nonetheless, it is likely that some word regarding the contents of the trial would have circulated due to the number of servants, soldiers, and opponents involved in the trial, and the charge would have been published at Jesus's execution (and would presumably have related to the preceding trial). "Accordingly, despite the absence of court records, no extreme skepticism about the basic issue of the Roman trial is warranted . . . ."

The Roman law under which Pilate condemned Jesus to death is never specifically stated in the Gospels. However, the charge levied against Jesus is: he allegedly claimed to be King of the Jews. Under Roman law, such a claim could constitute sedition. Specifically, most scholars believe it would be an offense against the Lex Iulia de maiestate, which "covered slander of the emperor and his family." Of course, just because Jesus's conduct may have been against the law does not mean that Jesus was put to death for breaking that law. Pilate may have simply had Jesus crucified for being an non-citizen troublemaker to help maintain order. In that case, the fact that Jesus may have happened to have broken Roman law would not have played strongly into Pilate's decision.