(3.10) L(w,txm)b(w,t,7r,m) -• |m|^" s L(l - w,£xm)b(l - w,£5r, m).
The existence of these weighting factors for r = 1, 2 and the bounds
(3.11) |a(s,7r,d)|
|d|c and |6(w,£,7r,m)|
|m|^ + e for S ( s ) , % ) 3/2
will be shown in the following section.
From (3.9),(3.10),(3.11) and the Phragmen-Lindelof principle, we deduce the
convexity bounds
(3.12) (1 -
IT x Xd)a(s, TT, d)
for K(s) - \
(3.13) ( l - ^ )
f c
L ( ^ , ^ X m ) K ^ ^ ^ ^ )
H for R(s) - |
where / is the order of the pole of L(s, TT x \d) at 5 = 1 and k is the order of the
pole of L(w, £,Xm) at w = 1. (Such poles occur only if TT is non-cuspidal with central
character \d or if £ = Xm-) Thus by absolute convergence, the representation
(3.7) of Z(s,w) defines an analytic function for 5ft(s) T-,$1(W) 4 and the
representation (3.8) is analytic for K(w;) —|,5ft(s) 2. Let X be the union of
these two regions. Then X is a connected tube domain. Let G be the finite group
of transformations of
generated by
(3.14) (s, u) (1 s, 10 + r(s ^)) and (s, to) H- (5 + w | , 1 it;)
As indicated in Section 3.1, the double Dirichlet series Z(s,w) has an invariance
with respect to this group G. Moreover, the tube domain X contains the comple-
ment of a compact subset of a fundamental domain for the action of G on C2.
Therefore the union of the translates of X by G is ft, say, a connected tube domain
which is the complement of a compact subset of
It follows that we can ana-
lytically continue Z(s,w) to the set SI, and in fact, P{s,w)Z(s,w) is holomorphic
on Q, where P(s,w) is a finite product of linear terms which clear the translates
of the possible polar lines s = l,w = 1 of Z(s,w). We now apply Theorem 3.2 to
analytically continue Z(s,w) to C2. A similar argument is presented elsewhere in
this volume in [9], Section 1, and the reader may wish to see the figure illustrating
it there.
For example, let TT be an automorphic representation of GL(3) with trivial
central character. The group G is dihedral of order 12. In [17] it is shown that
w(w - 1)(35 + w - 5/2)(3s -f 2w - 3)(3s + w - 3/2) x bad prime factor x Z(s,w)
has an analytic continuation to C2.
Similarly, the multiple Dirichlet series (with suitable weight factors) corre-
sponding to GL(1) x GL(1) and GL(1) x GL(2) (resp. GL(1) x GL(1) x GL(1))
given by (2.1) meromorphically continue to C3 (resp. C4) with a finite number of
polar hyperplanes. The weight factors needed to make the heuristic rigorous (i.e.
to show that a sum of Euler products in the Si is also a sum of Euler products in
w) are once again unique.
Though the heuristics are easiest to explain over Q, we emphasize that the
method works over a general global field [31],[32]. To do so, one must pass to
a ring of 5-integers that has class number one, and look at a finite dimensional
vector space of multiple Dirichlet series. This space is stable under the functional
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