Determination of GL(3) Cusp Forms by Central Values of Quadratic Twisted L-Functions

Abstract

Abstract Let $\phi $ and $\phi ^{\prime}$ be two $\operatorname {GL}(3)$ Hecke–Maass cusp forms. In this paper, we prove that $\phi =\phi ^{\prime} \textrm {or }\widetilde {\phi ^{\prime}}$ if there exists a nonzero constant $\kappa $ such that $L(\frac {1}{2},\phi \otimes \chi _{8d})=\kappa L(\frac {1}{2},\phi ^{\prime}\otimes \chi _{8d})$ for all positive odd square-free positive $d$. Here, $\widetilde {\phi ^{\prime}}$ is dual form of $\phi ^{\prime}$ and $\chi _{8d}$ is the quadratic character $(\frac {8d}{\cdot })$. To prove this, we obtain asymptotic formulas for twisted 1st moment of central values of quadratic twisted $L$-functions on $\operatorname {GL}(3)$, which will have many other applications.