Hilchos Bikkurim 11:12

If a kohein has a son who is a chalal* and the father dies within 30 days of the birth, the son must redeem himself because the father didn’t acquire the redemption money (i.e., he could just “give” the money to himself, metaphorically speaking). If the kohein father dies more than 30 days after the birth, the son need not redeem himself because the father acquired the redemption money.

Hilchos Bikkurim 11:13

Let’s say that a maidservant was freed or a non-Jewish woman converted while pregnant and subsequently gave birth. The son must redeem himself since he was born Jewish, even though he was not conceived Jewish. This is derived from “the first issue of the womb in Israel” (a paraphrase of Exodus 13:2 and Numbers 3:12), and this child is the first issue of a womb in Israel. If we don’t know whether the son was born before or after the woman converted, we act based on the rule that the burden of proof is on the one who wants to take property from another (i.e., the kohein must prove that the son was born after conversion if he wants to collect the redemption money).

*A non-kohein because he is the product of a relationship prohibited to kohanim.