S}\alpha_{\cE_v}^\#(\phi_v,\phi_v)^{-1} \\
& \sum_{E\in X(D,\pi,\cE_S)} \frac{ L(1,\eta)^2 \, |\cP_E(\phi)|^2\, \mathcal{D}_E^S(\pi,s)}{N(\mathfrak{f}_\eta^S)^{s-1}}\end{aligned}$$ (Theorem \[thm:20200105t1\]). Here, $\phi=\otimes_v \phi_v$ is a suitably chosen cusp form in $\pi$, $\mathcal{D}_E^S(\pi,s)$ is a function defined as a certain infinite product, and $X(D,\pi,\cE_S)$ denotes the set of $E\in X(D)$ whose $v$-component coincides with $\cE_v$ for $v\in S$ such that $\pi$ is $E^\times$-distinguished. For other unexplained notation, see § \[sec:w0504\], § \[sec:fac\] and § \[sec:exp\]. Together with the assumptions of Theorem \[thm:main\] and an element $\cE_S'=\prod_{v\in S}\cE_v'$ such that $X(D,\pi,\cE_S')$ is not empty, the assertion follows from the following observation and comparison of the order of the pole at $s=1$ of the both sides of the above equation:

-   The global zeta function $Z(\Phi,\phi,\overline{\phi},s)$ is holomorphic at $s=1$.

-   $\zeta_F^S(2s-1)$ has a simple pole at $s=1$.

-   With suitable choices of $\Phi$ and $\phi$, the local zeta function $Z_{\cE_v'}(\phi_v,\Phi_v,s)$ is entire and does not have zero at $s=1$ for each $v\in S$, and $Z_{\cE_v}(\phi_v,\Phi_v,s)=0$ if $\cE_v\neq \cE_v'$.

-   The infinite product defining $\mathcal{D}_E^S(\pi,s)$ converges absolutely for $\mathrm{Re}(s)>1$ and $\mathrm{Re}(\mathcal{D}_E^S(\pi,s))$ tends to $+\infty$ when $s\rightarrow 1+0$.

See the proofs of Theorems \[thm:nonvanishing423\] and \[thm:infinite\] for the detail.

In order to refine our result to get a generalization of Theorem \[thm:FH\], 