The Whittaker-Shintani functions for symplectic groups

In this note, we give a formula for the Whittaker-Shintani functions for the p-adic symplectic groups, which is a generalization of the Zonal spherical functions and Whittaker functions. We then use the formula to give an alternative proof of a conjecture given by T.Shintani on the unramified calculation of L-function for $Sp_{2n}\times GL_{1}$.

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