have $\ve(1/2, \pi'\otimes\chi)=1$ for some $\chi\in\Psi(S;\chi_0)$. Then there are infinitely many $\chi\in\Psi(S;\chi_0)$ such that $L(1/2, \pi'\otimes\chi)\neq0$.

In the proof, Waldspurger used in an essential way the result of Flicker on the Shimura lifting from $\GL_2$ to a certain double cover of $\GL_2$. Instead of it, the argument of Friedberg and Hoffstein is based on a technique of analytic number theory. They studied poles of a Dirichlet series with coefficients given by $L$-functions.

These theorems on special values of $L$-functions can be restated in terms of nonvanishing of periods. In order to do that, we recall another theorem of Waldspurger from [@Wal4]. Let $X(E)$ denote the set of quatenion algebras over $F$ which $E$ embeds as an $F$-algebra. For $D\in X(E)$, we set $D_\A=D\otimes_F\A$. For an irreducible cuspidal automorphic representation $\pi$ of $D_\A^\times$, there is a unique irreducible automorphic representation $\pi'=\operatorname{JL}(\pi)$ of $\GL_2(\A)$ such that for almost all places $v$ of $F$ at which $D$ splits, $\pi_v$ is isomorphic to $\pi'_v$.

\[thm:W2\] (1) For $D\in X(E)$, let $\pi$ be an irreducible cuspidal automorphic representation of $D_\A^\times$ with trivial central character such that $\pi'=\operatorname{JL}(\pi)$ is cuspidal. If $\pi$ is $E^\times$-distinguished, then $L(1/2, \pi')L(1/2, \pi'\otimes\eta)\neq0$.

\(2) Let $\pi'$ be an irreducible cuspidal automorphic 