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By Diaconu A., Garrett P

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87 (1997), 343–353. [B] A. Borel, Introduction to automorphic forms, Algebraic Groups and Discontinuous Subgroups, Proc. Symp. Pure Math. 9, AMS, Providence, 1966, pp. 199–210. [BM1] R. W. Bruggeman and Y. Motohashi, Fourth power moment of Dedekind zeta-functions of real quadratic number fields with class number one, Funct. Approx. Comment. Math. 29 (2001), 41–79. [BM2] R. W. Bruggeman and Y. Motohashi, Sum formula for Kloosterman sums and fourth moment of the Dedekind zeta-function over the Gaussian number field, Funct.

Godement, The spectral decomposition of cuspforms, Algebraic Groups and Discontinuous Subgroups, Proc. Symp. Pure Math. 9, AMS, Providence, 1966, pp. 225–234. [G1] A. Good, The square mean of Dirichlet series associated with cusp forms, Mathematika 29 (1982), 278–295. [G2] A. Good, The Convolution method for Dirichlet series, The Selberg trace formula and related topics, (Brunswick, Maine, 1984) Contemp. Math. 53, American Mathematical Society, Providence, RI, 1986, pp. 207–214. [GR] I. S. Gradshteyn and I.

While the details of this computation are given below, we also cite [W2], Chapter VII, for standard facts about the Tate-Iwasawa theory of zeta integrals. The global Mellin transform of W E factors |a|v Ws,Eχ J a 0 0 1 da = ν kν× |a|vν Ws,Eχ, ν a 0 0 1 da To compute this, we cannot simply change the order of integration, since this would produce a divergent integral along the way. Instead, we present the vectors ην in a different form. Let Φν be any Schwartz function on kν2 invariant under Kν (under the obvious right action of GL2 ), and put ην (g) = χν (det g) | det g|sν · kν× χ2ν (t) |t|2s ν · Φν (t · e2 · g) dt where e2 = e2, ν is the second basis element in kν2 .

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