  According to the Birch and Swinnerton-Dyer conjectures, if A/Q is an abelian
variety then its L-function must capture substantial part of the arithmetic
properties of A. The smallest number field L where A has all its endomorphisms
defined must also have a role. This article deals with the relationship between
these two objects in the specific case of modular abelian varieties A_f/Q
associated to weight 2 newforms for the modular group Gamma_1(N). Specifically,
our goal is to relate the order of L(A_f/Q,s) at s = 1 with Euler products
cropped by the set of primes that split completely in L. The results we obtain
for the case when f has complex multiplication are complete, while in the
absence of CM, our results depend on the rate of convergence in Sato-Tate
distributions.
