  Given a pure motive $M$ over $\mathbb{Q}$ with a multilinear algebraic
structure $\mathsf{s}$ on $M$, and given a representation $V$ of the group
respecting $\mathsf{s}$, we describe a functorial transfer $M^V$. We formulate
a criterion that guarantees when the two periods of $M^V$ are equal. This has
an implication for the critical values of the $L$-function attached to $M^V.$
The criterion is explicated in a variety of examples such as: tensor product
motives and Rankin-Selberg $L$-functions; orthogonal motives and the standard
$L$-function for even orthogonal groups; twisted tensor motives and Asai
$L$-functions.
