h$$ defines a non-zero linear form on $\pi$. Here, $Z$ denotes the center of $\GL_2$.

Fix a finite set $S$ of places of $F$. For $\xi\in F^\times$, let $\chi_\xi$ denote the quadratic character $(\,\cdot, \xi)_\A$ of $\A^\times$, where $(\cdot,\cdot)_\A$ is the product of local Hilbert symbols. The next theorem is [@Wal4 Théorèm 4].

\[thm:W1\] Let $\pi'$ be an irreducible cuspidal automorphic representation of $\GL_2(\A)$ with trivial central character. Take a positive real number $\de_v$ for each $v\in S$. If we have $\ve(1/2, \pi')=1$, there exists $\xi\in F^\times$ such that

-   For any $v\in S$, $|\xi-1|_v<\de_v$.

-   $L(1/2, \pi'\otimes\chi_\xi)\neq0$.

As noted in the introduction of [@GI], this theorem plays an important role in the argument of Waldspurger to complete the full description of the automorphic discrete spectrum of $\mathrm{Mp}_2$.

Friedberg and Hoffstein generalized Theorem \[thm:W1\] to the following result. Fix a quadratic character $\chi_0$ of $\A^\times/F^\times$. Let $\Psi(S; \chi_0)$ denote the set of quadratic characters $\chi$ of $\A^\times/F^\times$ such that $\chi_v=\chi_{0,v}$ for all $v\in S$. For each non-trivial $\chi\in\Psi(S;\chi_0)$, let $E_\chi$ be the corresponding quadratic extension of $F$ via the global class field theory.

\[thm:FH\] Let $\pi'$ be an irreducible cuspidal automorphic representation of $\GL_2(\A)$ which is self-dual. Suppose that we 