https://en.wikipedia.org/wiki/Nabonidus

Babylon’s last native king was Nabonidus, who reigned from 556 to 539 BC. Nabonidus’s rule was ended through Babylon being conquered by Cyrus the Great of the Achaemenid Empire. Though early Achaemenid kings continued to place importance on Babylon and continued using the title ‘king of Babylon’, later Achaemenid rulers being ascribed the title is probably only something done by the Babylonians themselves, with the kings themselves having abandoned it. Babylonian scribes continued to recognise rulers of the empires that controlled Babylonia as their kings until the time of the Parthian Empire, when Babylon was gradually abandoned. Though Babylon never regained independence after the Achaemenid conquest, there were several attempts by the Babylonians to drive out their foreign rulers and re-establish their kingdom, possibly as late as 336/335 BC under the rebel Nidin-Bel.