Critical Values of Motivic L-Functions

Let M be a self-dual motive of odd weight whose L-function is automorphic. We show that certain polynomials generated by the special values of L(s,M) have
all of their zeros equidistributed on the unit circle, provided that one of the weight, rank, or global conductor of M is sufficiently large. This generalizes the recent
proof of the "Riemann hypothesis" for period polynomials of newforms by Jin, Ma, Ono, and Soundararajan. Our results impose conditions on the size of the critical
values of L(s,M), which have arithmetic significance in the context of the Bloch-Kato conjecture.