On the archimedean Euler factors for spin L–functions

The archimedean Euler factor in the completed spin L–function of a Siegel modular form is computed. Several formulas are obtained, relating this factor to the recursively defined factors of Andrianov and to symmetric power L–factors for GL(2). The archimedean ε–factor is also computed. Finally, the critical points of certain motives in the sense of Deligne are determined. Introduction Let f be a classical holomorphic Siegel modular form of weight k and degree n, assumed to be an eigenform of the Hecke algebra. Andrianov [An] has associated to f an L–function Zf (s) as an Euler product over all finite primes, where the Euler factor at p is a polynomial in p−s of degree 2. This is called the spin L–function of f since it is attached to the 2–dimensional spin representation of the L–group Spin(2n+ 1,C) of the underlying group PGSp(2n), see [AS]. A serious problem is that of obtaining the analytic continuation and a functional equation for Zf (s). The case n = 1 is the classical Hecke theory, the case n = 2 was done by Andrianov in [An]. Beyond that, very little is presently known. To obtain smooth functional equations, the partial L–function Zf (s) has to be completed with an Euler factor at the archimedean prime. For example, for n = 1, the function L(s, f) = (2π)Γ(s)Zf (s) has the functional equation L(s, f) = (−1)L(k − s, f). For n = 2, the definition L(s, f) = (2π)−2sΓ(s)Γ(s− k+2)Zf (s) leads to the functional equation L(s, f) = (−1)L(2k− 2− s) proved in [An]. The purpose of this paper is to give formulas for the archimedean Euler factor in any degree. Automorphic representation theory provides the recipe to compute this factor. Let Πk be the archimedean component of the automorphic representation of PGSp(2n,A) attached to the eigenform f (see [AS]). By the local Langlands correspondence over R, there is an associated local parameter φ : WR → Spin(2n + 1,C), where WR is the real Weil group. Let ρ be the spin representation. Then the factor we are looking for is L(s, Πk, ρ) = L(s, ρ ◦ φ), where on the right we have the L–factor attached to a finite-dimensional representation of the Weil group. The formulas we will thus obtain coincide for n = 1 and n = 2 with the Γ–factors from above, except for some constants. Note however that we are working with the automorphic normalization that is designed to yield a functional equation relating s and 1 − s. To compare with the classical formulas, we have to make a shift in the argument s. At the end of the paper [An] Andrianov gives another definition of an archimedean Euler factor by a recursion formula. It turns out that for n ≥ 3 this definition leads to a factor that is different from ours.