When was the temple rebuilt after the Babylonian exile?

Ezra 6:15 gives the completion of the temple after the Babylonian exile as 3 Adar, and the sixth year of Darius. > And this house was finished on the third day of the month Adar, which > was in the sixth year of the reign of Darius the king. Josephus, Antiquities 11.107 gives the completion of the temple after the Babylonian exile as 23 Adar in the *ninth* year of Darius. > Now the temple was built in seven years’ time. And in the ninth year > of the reign of Darius, on the twenty-third day of the twelfth month, > which is by us called Adar, but by the Macedonians Dystrus, the > priests, and Levites, and the other multitude of the Israelites, > offered sacrifices, as the renovation of their former prosperity after > their captivity, and because they had now the temple rebuilt [. . .] The general position, academically, appears to favor the year 516/515 BCE. This is clearly deduced according to Ezra rather than Josephus, since Darius I reigned from 522–486 BCE, his sixth year of reign necessarily beginning in 516 BCE. Given two points of logical understanding, that 1) Josephus claimed to be in possession of biblical materials taken from the temple after the siege in 70 CE, and was therefore potentially accessing older copies of the scriptures than anything we have today, and 2) that the numbers in our current text in other places don't always add up correctly, particularly concerning time periods following the exodus out of Egypt . . . would anyone have any insight into other sources on the rebuilding of the temple that might clarify the year? The relevance of the question is part of a larger study into the priestly courses. I've backdated them from the destruction of the temple in 70 CE per the Talmud, bTa'an 29a and the course of Jehoiarib on the 9th of Ab. But to "check my work" as they say, I'm trying to determine, as best as possible, the precise starting point when the courses were initially reinstituted in Ezra 6:18 immediately following the rededication of the temple. > And they set the priests in their divisions, and the Levites in their > courses, for the service of God, which is at Jerusalem; as it is > written in the book of Moses. Determining the correct year will allow me to count forward to verify the rotation. Assuming it wasn't broken in the time of the Maccabees, if I can determine the correct year, whether it's the sixth or ninth of Darius, counting forward should result in a match for Jehoiarib on the 9th of Ab if everything is on the up and up. Any additional material anyone can suggest is always especially welcomed.