Incidentally, there is speculation that the Gospel of Luke depicts the birth of Christ as having occured about that time. That would explain the crowding in Bethlehem, and of course, the shepards in the fields.
Zacharias served as a high priest and based on his tribe, we know when he served John would have been born on Pesach. Most Jews believed that Elijah would come at Factor in when Miriam visited her cousin Nine months back from Sukkot is Chanukah where the light entered the temple. Yah'shua's birth on Sukkot is supported by Elizabeth's pregnancy of John the Immerser.
shalom b'shem Yah'shua
The time sequence is outlined by the Holy Word of Elohim in Luke 1 with Zacharias.
and when he was struck dumb and when John was conceived.
Pesach to announce the coming of the Messiah
Elizabeth, timing of Yah'shua's birth can be ascertained.