## The Generalized Whittaker Functions for $Sp(2, \Bbb R)$ and the Gamma Factor of the Andrianov $L$-function

J. Math. Sci. Univ. Tokyo
Vol. 7 (2000), No. 2, Page 241--295.

Miyazaki, Takuya
The Generalized Whittaker Functions for $Sp(2, \Bbb R)$ and the Gamma Factor of the Andrianov $L$-function
We study the archimedean generalized Whittaker functions for the generalized principal series and the large discrete series of the real symplectic group of degree 2. Using gradient type differential operators, which was introduced by Schmid, we give a system of differential equations which is satisfied by a Whittaker function. We study this system, and give the Mellin transform of its solution. We apply the result to a study of Andrianov's spinor $L$-function for a non-holomorphic Siegel modular form via Rankin-Selberg integral with an explicitly described archimedean factor.