In the time of Herod king of Judea there was a priest named Zechariah, who belonged to the priestly division of Abijah; his wife Elizabeth was also a descendant of Aaron. 

I seem to remember David Pawson commenting that the Christmas story is about two babies, not one, and Luke shows, from the beginning of his gospel, how the lives and ministries of John the Baptist and Jesus overlapped.

‘The careful structuring and interweaving of the stories indicates that John was a significant person whose birth was part of God’s plan and that Jesus was an even greater person than his predecessor. The story contains many echoes of the OT which show that God’s new acts were in harmony with his earlier mighty acts for his people and also in fulfilment of prophecy. The various supernatural occurrences recorded also mark out the two infants as God’s servant and Son respectively.’ I.H. Marshall: ‘New Bible Commentary’, p.981.

At one time, many scholars questioned just how accurate a historian Luke was. But over the years, the good doctor has been repeatedly vindicated by numerous archaeological findings. Having stated in the opening verses his commitment to producing an accurate account, here he is earthing John’s story in a particular historical moment.