Period Relations for Standard $L$-functions of Symplectic Type

This article is to understand the critical values of $L$-functions $L(s,Piotimes chi)$ and to establish the relation of the relevant global periods at the critical places. Here $Pi$ is an irreducible regular algebraic cuspidal automorphic representation of $mathrm{GL}_{2n}(mathbb A)$ of symplectic type and $chi$ is a finite order automorphic character of $mathrm{GL}_1(mathbb A)$, with $mathbb A$ is the ring of adeles of a number field $mathrm k$.