easy to prove the existence of such a family $\{\cE_v\}$ in general. Conversely, the existence of such $\{\cE_v\}_v$ follows immediately from Theorem \[thm:main\].

Our method to prove Theorem \[thm:main\] is to use prehomogeneous zeta functions with period integrals. We will show in Theorem \[thm:nonvanishing423\] that if our zeta functions have a pole, then there exists such a quadratic extension $E_0$. The proof of this fact is not so long. Hence, if one combines Theorem \[thm:FH\] with Theorem \[thm:nonvanishing423\], then one obtains a short proof of Theorem \[thm:main\]. On the other hand, we will give an alternative proof in Theorem \[thm:infinite\] by the explicit formula of the zeta functions, that is, Theorem \[thm:main\] is proved by our zeta functions without using Theorem \[thm:FH\]. In the following subsections, we explain the zeta functions, the proofs and possibilities of further applications.

Prehomogeneous zeta functions with period integrals
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Our main tool is the prehomogeneous zeta functions with period integrals which were introduced by F. Sato in [@Sato]. Besides the application to the study of period integrals, this paper has contribution to the theory of prehomogeneous zeta functions with period integrals. In this subsection, we give an overview of prior research on this topic.

