Luke quotes Leviticus 12:6 applying it to Mary=she had an issue of blood:
Luke 2:22 And when the days of her purification according to the law of Moses were accomplished,
Leviticus 12:5 But if she shall bear a maid child, she shall be unclean two weeks, **according to the custom of her monthly courses. ** And she shall remain in the blood of her purification sixty-six days. 6 And when the days of her purification are expired, for a son, or for a daughter, she shall bring to the door of the tabernacle of the testimony, a lamb of a year old for a holocaust, and a young pigeon or a turtle for sin: and shall deliver them to the priest. 7 Who shall offer them before the Lord, and shall pray for her: and so **she shall be cleansed from the issue of her blood. ** This is the law for her that beareth a man child or a maid child.
The impurity is identified with and expressly defined as the issue of blood during pregnancy. If there were no issue of blood there would be no impurity and therefore no need to sacrifice turtledoves.
When Mary sacrificed two turtledoves, one as an offering for sin, it was for her sin of defiling God’s dwelling by having such while in atonement with Him.
Douay Rheims Leviticus 15: 29 And on the eighth day she shall offer for herself to the priest, two turtles, or two young pigeons, at the door of the tabernacle of the testimony: 30 And he shall offer one for sin, and the other for a holocaust, and he shall pray for her before the Lord, and for the issue of her uncleanness. 31 You shall teach therefore the children of Israel to take heed of uncleanness, that they may not die in their filth, when they shall have defiled my tabernacle that is among them.
The turtledove cleanses sin from her so she can be allowed back into God’s presence, His Tabernacle-dwelling: = atonement restored.
The other turtledove is for restored communion like that experienced at a “family meal” symbolizing atonement has been restored, sin cleansed from her that enables her to be in God’s presence.