In his book, "Introduction to the arithmetic theory ..." (Ch 7, section 8) Shimura constructs the Zeta function of an abelian variety with CM and expresses it as a product of L-functions.

Since I found his proof of the main theorem of complex multiplication for elliptic curves very messy, (along with other proofs - and it's not clear that this one can be made simpler) I'd like to ask:

Are there other sources where the zeta function of an abelian variety is constructed (elliptic curve case is in Lang's Elliptic functions), or would you suggest learning this from Shimura?