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