1 Chr 24:19 These were the orderings of them in their service to come into the house of the LORD, according to their manner, under Aaron their father, as the LORD God of Israel had commanded him.

Now each one of the 24 "courses" of priests would begin and end their service in the Temple on the Sabbath, a tour of duty being for one week (2 Chr 23:8, 1 Chr 9:25). On three occasions during the year, all the men of Israel were required to travel to Jerusalem for festivals of the Lord, so on those occasions all the priests would be needed in the Temple to accommodate the crowds. Those three festivals were Unleavened Bread, Pentecost, and Tabernacles (Deut 16:16).

The Yearly Cycle of Service in the Temple.

The Jewish calendar begins in the spring, during the month of Nisan, so the first "course" of priests, would be that of the family of Jehoiarib, who would serve for seven days. The second week would then be the responsibility of the family of Jedaiah. The third week would be the feast of Unleavened Bread, and all priests would be present for service. Then the schedule would resume with the third course of priests, the family of Harim. By this plan, when the 24th course was completed, the general cycle of courses would repeat. This schedule would cover 51 weeks or 357 days, enough for the lunar Jewish calendar (about 354 days). So, in a period of a year, each group of priests would serve in the Temple twice on their scheduled course, in addition to the 3 major festivals, for a total of about five weeks of duty.

The Conception of John the Baptist.

Now back to Zacharias, the father of John the Baptist. Luke 1:23 And it came to pass, that, as soon as the days of his ministration were accomplished, he departed to his own house. Luke 1:24 And after those days his wife Elisabeth conceived, ...

Beginning with the first month, Nisan, in the spring (March-April), the schedule of the priest's courses would result with Zacharias serving during the 10th week of the year. This is because he was a member of the course of Abia (Abijah), the 8th course, and both the Feast of Unleavened Bread (15-21 Nisan) and Pentecost (6 Sivan) would have occurred before his scheduled duty. This places Zacharias' administration in the Temple as beginning on the second Sabbath of the third month, Sivan (May-June).

Having completed his Temple service on the third Sabbath of Sivan, Zacharias returned home and soon conceived his son John. So John the Baptist was probably conceived shortly after the third Sabbath of the month of Sivan.

Now the reason that the information about John is important, is because according to Luke, Jesus was conceived by the Holy Spirit in the sixth month of Elisabeth's pregnancy: Luke 1:24 And after those days his wife Elisabeth conceived, and hid herself five months, saying,  Luke 1:25 Thus hath the Lord dealt with me in the days wherein he looked on me, to take away my reproach among men.
Luke 1:26 And in the sixth month the angel Gabriel was sent from God unto a city of Galilee, named Nazareth, Luke 1:27 To a virgin espoused to a man whose name was Joseph, of the house of David; and the virgin's name was Mary.

Note that verse 26 above refers to the sixth month of Elisabeth's pregnancy, not Elul, the sixth month of the Hebrew calendar, and this is made plain by the context of verse 24 and again in verse 36: Luke 1:36 And, behold, thy cousin Elisabeth, she hath also conceived a son in her old age: and this is the sixth month with her, who was called barren.

Now working from the information about John's conception late in the third month, Sivan, and advancing six months, we arrive late in the 9th month of Kislev (Nov-Dec) for the time frame for the conception of Jesus. It is notable here that the first day of the Jewish festival of Hanukkah, the Festival of Lights, is celebrated on the 25th day of Kislev, and Jesus is called the light of the world (John 8:12, 9:5, 12:46). This does not appear to be a mere coincidence. In the book of John, Hanukkah is called the feast of dedication (John 10:22). Hanukkah is an eight day festival, celebrating the relighting of the menorah in the rededicated Temple, which according to the story, stayed lit miraculously for eight days on only one day's supply of oil.

Based on a conception shortly after the third Sabbath of the month of Sivan, projecting forward an average term of about 10 lunar months (40 weeks), we arrive in the month of Nisan. It would appear that John the Baptist may have been born in the middle of the month, which would coincide with Passover and the Feast of Unleavened Bread. It is interesting to note, that even today, it is customary for the Jews to set out a special goblet of wine during the Passover Seder meal, in anticipation of the arrival of Elijah that week, which is based on the prophecy of Malachi:

Mal 4:5 Behold, I will send you Elijah the prophet before the coming of the great and dreadful day of the LORD: Jesus identified John as the "Elijah" that the Jews had expected: Mat 17:10 And his disciples asked him, saying, Why then say the scribes that Elias must first come? Mat 17:11 And Jesus answered and said unto them, Elias truly shall first come, and restore all things. Mat 17:12 But I say unto you, That Elias is come already, and they knew him not, but have done unto him whatsoever they listed. Likewise shall also the Son of man suffer of them. Mat 17:13 Then the disciples understood that he spake unto them of John the Baptist. The angel that appeared to Zacharias in the temple also indicated that John would be the expected "Elias": Luke 1:17 And he shall go before him in the spirit and power of Elias, to turn the hearts of the fathers to the children, and the disobedient to the wisdom of the just; to make ready a people prepared for the Lord.

So then, the Feast of Unleavened Bread begins on the 15th day of the 1st month, Nisan, and this is a likely date for the birth of John the Baptist, the expected "Elijah".

Since Jesus was conceived six months after John the Baptist, and we have established a likely date for John's birth, we need only move six months farther down the Jewish calendar to arrive at a likely date for the birth of Jesus. From the 15th day of the 1st month, Nisan, we go to the 15th day of the 7th month, Tishri. And what do we find on that date? It is the festival of Tabernacles! The 15th day of Tishri begins the third and last festival of the year to which all the men of Israel were to gather in Jerusalem for Temple services. (Lev 23:34)

Conclusion