The sixth century abbot, Dionysius Exiguus, created the calendar based on the Christian Era. Before that time, there were no BC or AD years. Instead, years were based on the reigns of kings or emperors e.g. the fifteenth year of Tiberius Caesar. There were two methods of recording those so called regnal years.
Ezra was a scribe who was skilled in such matters as the reigns of kings. He consistently used accession year reckoning to record specific years in the reigns of Persian emperors. Under that system, the accession year, most often a partial year, wasn’t included in a king’s regnal years.
Ezra’s book began in the first year of Cyrus king of Persia. If you've read my article entitled The Conquest of Babylon, you know that Cyrus captured Babylon without a battle. On October 29, 539 BC, he entered the city’s gates and, with thunderous applause, its inhabitants acknowledged him as emperor. His first regnal year, as ruler of Babylon, was 538/537 BC according to accession year reckoning.
Artaxerxes reigned from 465-424 BC. His decree that sent Ezra to Jerusalem was proclaimed in his seventh regnal year. That year was 458/457 BC according to accession year reckoning based on Judah's civil calendar which celebrated an autumn new year.
The reign of Darius was a little more complicated. I like the chronology presented by Yamauchi, author of “Persia and the Bible.” Shortly before Cambyses' death, Gaumata the usurper began his revolt on March 11, 522 BC. By mid-April, he was recognized as emperor in Babylonia and by July 1, he was recognized as emperor throughout the entire Persian Empire. On September 29, his masquerade ended and he was slain by Darius the Great.
Like Gaumata, it took a few weeks before Darius was recognized as emperor. He wasn’t the obvious heir and had to subdue those who challenged his claim to the throne. His accession occurred in the Jewish year 522/521 BC. Ezra said that Zerubbabel’s temple was completed in the sixth year of Darius the Great. Using accession year reckoning, we arrive at 516/515 BC. In fact, Yamauchi says the second temple was finished on March 12, 515 BC.
Inclusive reckoning was the popular method of recording time when the Gospels were written. Under that system, partial years and partial days were counted as complete years and complete days. For example, the time between Christ’s death and resurrection was about forty hours. He died at 3 PM on Good Friday and rose from the dead at dawn on Easter Sunday. However, it was recorded as three days. That’s because the part days on Friday and Sunday were counted as full days.
Luke used inclusive reckoning to record the year of Christ’s baptism. He was baptized in the fifteenth year of Tiberius Caesar. Tiberius’ accession year was 14 AD. That partial year was considered his first regnal year. Hence his fifteenth regnal year was 28 AD, the calendar year when Jesus was baptized.