we should study the local zeta functions at “bad places” in more detail. To be more precise, what is needed for such a refinement is nonvanishing of the gamma factors which appear in the local functional equations proved by Wen-Wei Li [@Li4]. In § \[sec:20200424\], we will discuss the prospect of the future works in this direction. A detailed study on analytic properties of local zeta functions provides us quantitative information. For example, from the study of the gamma factors, we might be able to obtain convexity bound of $\xi(s,D,\phi,\cE_S)$, see [@SS Theorem 2 (iii)]. As another application of our global and local zeta functions, probably we can use the Dirichlet series $\xi(s,D,\phi,\cE_S)$ to study the mean value theorem for $\lim_{X\to\inf}X^{-1}\sum_{E\in X(D,\pi,\cE_S),\, N(\mathfrak{f}_\eta^S)^{1/2}<X}L(1,\eta)^2 \, |\cP_E(\phi)|^2$, see [@KY1] and [@Taniguchi2].

The following is the structure of the paper. In the next section, we recall the structure theory of the prehomogeneous vector spaces we consider and Waldspurger’s formula for toric periods. In § \[sec:globalzeta424\], we recall the notion of the prehomogeneous zeta functions with period integrals and show their basic properties such as absolute convergence, meromorphic continuation and the global functional equation. From the analysis of the principal part and the explicit 