Shimura and Taniyama proved that if A is a potentially CM abelian variety over a number field F with CM by a field K linearly disjoint from F, then there is an algebraic Hecke character λ A of F K such that L(A/F, s) = L(λ A , s). We consider a certain converse to their result. Namely, let A be a potentially CM abelian variety appearing as a factor of the Jacobian of a curve of the form y e = γx f + δ. Fix positive integers a and n such that n/2 < a ≤ n. Under mild conditions on e, f, γ, δ, we construct a Chow motive M , defined over F = Q(γ, δ), such that L(M/F, s) and L(λ a A λ n−a A , s) have the same Euler factors outside finitely many primes.