Chow motives associated to certain algebraic Hecke characters

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 ${\lambda }_A$ of $K$ such that $L(A\mathrm{/}F,s)=L({\lambda }_A,s)$. 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 $y^e=\gamma x^f+\delta$. Fix positive integers $a$ and $n$ such that $n\mathrm{/}2<a\le n$. Under mild conditions on $e, f, \gamma, \delta$,weconstructaChowmotive$M$,definedover$F=$\mathbb{Q}$(\gamma,\delta)$,suchthat$L(M/F,s)$and$L(\lambda_A^a\bar{\lambda}_A^{n-a},s)$ have the same Euler factors outside finitely many primes.