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

Abstract: This article is to understand the critical values of $L$-functions $L(s,\Pi\otimes \chi)$ 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 $\mathrm k$.