Someone just asked me why I think that Luke introduces his genealogy in chapter 3, after Jesus’ baptism, rather than the beginning like Matthew does. I think the answer is simple: Jesus is being baptized into his priestly duties.
In the Levitical system, most of the washings prescribed are self-washes (Lev 14:8; 15:5, 7, 10). One of the major differences was that Aaron was baptized by Moses (Lev 8:6) in preparation for his new role as priest. The priest is a sin-bearer for his people (Leviticus 16), having to wash himself to be cleansed before he could be purified in place of the people. This makes sense of why Jesus would need to be baptized by John: someone else needed to wash the new high priest and Jesus would have to be purified before he could bear the sins of the people.
At his baptism, Jesus is declared to be the Son of God. This title has kingly connotations, but it also refers to the priests. The Levites took the place of the firstborn of Israel (Numbers 3:38-51). Similarly, Paul in Hebrews quotes Psalm 2:7 in reference to the coronation of the new High Priest in Heb 4:14 and calls the new High Priest “Jesus the Son of God” (5:15). Right after the baptism, Jesus begins his new ministry around the age of thirty. Thirty is the age that priests begin their work (Num 4:34-37, also Ezekiel).
That all being said, if Jesus is truly a priest, then a genealogy is completing fitting. We know that the Jews recorded genealogies for their priests to make sure that they traced back to Aaron (Ezra 2:61-63). The change here is that Jesus cannot trace his ancestry back to Levi: he traces his back to Judah. So, Luke traces the genealogy all the way back to Adam, the original priest. Jesus’ priestly ministry is like Adam’s: they are both to protect the Bride (Israel, Eve) by defeating the Serpent. Jesus does so in the next story: he outwits the Serpent who tries to force him to perform a coup for the kingdom (Luke 4:6-8).
For the exegete: much ink could be spilled discussing how Melchizedek’s ministry is the direct continuation of Adam’s.