arbitrary admissible representations satisfy certain local functional equations.

On the one hand the local theory is now established in general by Li, but on the other hand nothing is known about the global theory and a lot of open questions are raised in [@Li1 Chapter 8]. We mention that this paper answers those questions for specific prehomogeneous vector spaces. Firstly, he explained in [@Li1 Corollary 8.3.7] that the nonvanishing of the zeta functions implies the automorphic representation is distinguished. In this paper, we will show that the zeta function has a pole provided the central value of the $L$-function is nonzero, and obtain a sufficient condition for distinction. Secondly, the meromorphic continuation and the functional equation of the zeta functions conjectured in [@Li1 Definition 8.4.1] is obtained in Theorem \[thm:global\] and Corollary \[cor:funct\]. Lastly, we will obtain the essential factorization , of our zeta functions, which is an expression as a sum of Euler products of local zeta functions proposed in [@Li1 Definition 8.2.5]. This expression will be obtained from the Saito’s formula and Waldspurger’s formula and plays an important role in the application to the study of the period integrals. In addition, we explicitly compute the local zeta functions 