122 J.W. COGDELL, L-FUNCTIONS FOR GLnthe domain of integration also naturally factors we can decompose this last integral
into an Euler product and now write
w(s; w<p, w~,, )= II Wv(s; w'Pv> w~~' v),
v
where
wv(s; W"'v' W~~, <I>v) = ( W"'Jgv)W~)9v)<I>v(en9v)I det(gvW dgv,
}Nn(kv)\ GLn(kv)still with convergence for Re( s) > > 0 by the local gauge estimates. Once again
we see that the Euler factorization is a direct consequence of the uniqueness of the
Whittaker models.
Theorem 2.2. Let <p E V7l' and <p' E V7l'' cusp forms on GLn(A) and let E
S(An). Then the family of integrals I(s;<p,<p',) define meromorphic functions
of s, bounded in vertical strips away from the poles. The only possible poles are
simple and occur iff 7r ~ if' © I det lia with r:r real and are then at s = -fo and
s = 1 - iCY with residues as above. They satisfy the functional equation
I(s; <p, <p^1 , <I>)= I(l - s; W"', W~,, <i>).Moreover, for <p, <p^1 , and factorizable we have that the integrals are Eulerian and
we have
v
with convergence absolute and uniform for Re(s) >> 0.
We remark in passing that the right hand side of the functional equation also
unfolds as
I(l - s; cp, <p', <i>) = ( W"'(g)W~,(g)<i>(eng)I det(g)l^1 -s dg
}Nn(A)\ GLn(A)
=II wv(1 -s; w"', w~,, <i>)
vwith convergence for Re( s) < < 0.
We note again that if these integrals are not entire, then the residues give us
invariant pairings between the cuspidal representations and hence tell us structural
facts about the relation between these representations.