  Let $F$ be a number field and $D$ a quaternion algebra over $F$. Take a
cuspidal automorphic representation $\pi$ of $D_\mathbb{A}^\times$ with trivial
central cahracter. We study the zeta functions with period integrals on $\pi$
for the perhomogeneous vector space $(D^\times\times
D^\times\times\mathrm{GL}_2, D\oplus D)$. We show their meromorphic
continuation and functional equation, determine the location and orders of
possible poles and compute the residue. Arguing along the theory of Saito and
computing unramified local factors, the explicit formula of the zeta functions
is obtained. Counting the order of possible poles of this explicit formula, we
show that if $L(1/2, \pi)\neq0$, there are infinitely many quadratic extension
$E$ of $F$ which embeds in $D$, such that $\pi$ has nonvanishing toric period
with respect to $E$.

Number Theory