$x(\delta)$ means a representative element of a $H(F)$-orbit in $V^0(F)$ corresponding to $E$, and $\d \tilde{h}$ denotes a quotient measure on $H_{x(\delta)}^0(\A)\bsl H(\A)$. For other unexplained notation, see § \[sec:space\]. From this, we see that $\pi$ is $E^\times$-distinguished for at least one $E\in X(D)$ if the zeta function is not identically zero. By analyzing the principal part of our zeta functions (Theorem \[thm:global\]), we will see that if $L(1/2, \pi)\neq0$, we can take $\Phi$ and $\phi$ so that $Z(s, \Phi, \phi, \bar{\phi}, s)$ has a simple pole at $s=1/2$ under the conditions of Theorem \[thm:main\]. In particular, $\pi$ is $E^\times$-distinguished for at least one $E\in X(D)$.

To show the nonvanishing of infinitely many periods, we need more close analysis on our zeta functions. Applying the technique of essential factorization developed by Saito for ordinary prehomogeneous zeta functions and computing explicitly the unramified components of the Euler factor, we get $$\label{eq:0419ex}
Z(\Phi,\phi,\overline{\phi},s)=\sum_{\cE_S\in X(D_S)}   \prod_{v\in S} Z_{\cE_v}(\phi_v,\Phi_v,s)  \times   \xi(s,D,\phi,\cE_S),$$ where $S$ is a finite set of places of $F$ containing all “bad” places, $\cE_S=\prod_{v\in S}\cE_v$ in $X(D_S):=\prod_{v\in S}X(D_v)$, $Z_{\cE_v}(\phi_v,\Phi_v,s)$ is the local zeta function (cf. ) and $$\begin{aligned}
\xi(s,D,\phi,\cE_S):=& \frac{ 1 }{2 \, |\Delta_F|^4  \, c_F }  \frac{\zeta^S_F(2s-1) \, L^S(2s-1,\pi,\mathrm{Ad}) }{\zeta^S_F(2)^3}  \prod_{v\in 