Another birth dating piece is the rotation of the Priesthood. We know that the first course of the Priests served in the Temple at the time of the destruction in the second week of the 5th month in 70 AD.
We know that the 8th course served during the day of Atonement when Zacharias was serving in the Temple in 4 B.C.E.
Luk 1:5 There was in the days of Herod, the king of Judaea, a certain priest named Zacharias, of the course of Abia: and his wife was of the daughters of Aaron, and her name was Elisabeth.
We simply plot the courses through the years backwards from 70 AD and see that the 8th course would be serving during the 2nd week of the 7th month in 4 B.C.E.