Motives, Periods, and Functoriality

Given a pure motive M over Q with a multilinear algebraic structure s on M, and given a representation V of the group respecting s, we describe a functorial transfer MV. We formulate a criterion that guarantees when the two periods of MV are equal. This has an implication for the critical values of the L-function attached to MV. 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.