Exceptional poles of local $L$-functions for $GSp(4)$ with respect to split Bessel models

Abstract

Piateskii-Shapiro defined local $L$-factors $L^{PS}(s,\Pi ,\Lambda )$ attached to irreducible admissible representations $\Pi$ of the group $GSp(4)$ over local fields and Bessel models of $\Pi$ attached to Bessel data $\Lambda$. These local $L$-factors decompose into a product $L^{PS}(s,\Pi ,\Lambda )=L_{ex}^{PS}(s,\Pi ,\Lambda )L_{reg}^{PS}(s,\Pi ,\Lambda )$ of an exceptional and a regular $L$-factor. In this paper we compute the exceptional factors $L_{ex}^{PS}(s,\Pi ,\Lambda )$ for split Bessel models of $\Pi$.