[@Saito1]. To the best of the authors’ knowledge, this is the first example that the prehomogeneous zeta function with period integrals are used to study nonvanishing of specific period integrals.

Let us switch back to the setting of Theorem \[thm:main\]. Set $G=D^\times\times D^\times\times\GL_2$ and $V=D\oplus D$. The $F$-rational representation $\rho$ of $G$ on $V$ is defined by $$(x, y)\cdot\rho(g_1, g_2, g_3):=(g_1^{-1}xg_2, g_1^{-1}yg_2)g_3, 
\quad (g_1,g_2,g_3)\in G, \quad (x,y)\in V.$$ Then, the triple $(G, \rho, V)$ is an $F$-form of a prehomogeneous vector space. We write the regular locus of $V$ by $V^0$. Let $\pi$ be a cuspidal automorphic representation of $D_\A^\times:=(D\otimes\A)^\times$ with trivial central character. For cusp forms $\phi_1, \phi_2$ in $\pi$ and a Schwartz function $\Phi$ on $V(\A)$, we define the global zeta function with period integrals by $$Z(\Phi, \phi_1, \phi_2, s)=\int_{H(F)\bs H(\A)}|\omega(h)|^s\phi_1(g_1)\phi_2(g_2)\sum_{x\in V^0(F)}\Phi(x\cdot\rho(h))\d h.$$ Here, $H$ is the quotient of $G$ by the kernel of $\rho\,\colon G\rightarrow\GL(V)$, $\omega$ is the character attached to the fundamental relative invariant of $(G, \rho, V)$ and $(g_1, g_2, g_3)$ is a representative in $G$ of $h\in H$.

By the orbit decomposition in Proposition \[prop:orbits\], we obtain $$\label{eq:0418ef}
Z(\Phi,\phi_1,\phi_2,s)=\frac{1}{2}\sum_{E\in X(D)}\int_{H_{x(\delta)}^0(\A)\bsl H(\A)} \cP_E(\phi_1^{g_1})\, \cP_E(\phi_2^{g_2})   \,  |\omega( \tilde{h} )|^s \Phi(x(\delta)\cdot \tilde{h}) \, \d \tilde{h},$$ where $\phi_i^{g_i}=\pi(g_i)\phi_i$, 