By Diaconu A., Garrett P
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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 .