20 GAUTAM CHINTA, SOLOMON FRIEDBERG, AND JEFFREY HOFFSTEIN
it thus follows that
6(ra0m2, 1) = -c(rn0ml)xmo{p)
for all p\mi. Referring to the recursion relation (3.28) and combining this with the
above we see that in the case (3 = 1 we have now determined the first Q polynomial:
Q(w,p2) = c(p2)(l~p~w+p1'2w).
Computing the coefficient of p~2s one obtains from the left hand side of (3.26)
Combining this with the Hecke relation
c(p)2
=
c(p2)
-f 1 and the information
a(dod21 1) = ~Xd0{p)c(p) obtained above this reduces to
V
1
4- V
a(d^2)
(p,d0d?) = l l P2\d:
The right hand side of (3.26) is
\(p,d) = l p|dl V U V
Equating the above two expressions we obtain
a(d$d\) = 1
if p\\d\ and
a(d0d?) = l+pc(p 2 )
if p2\d\. Thus because of the recursion relations we have completely determined
the first P polynomial:
P(s,doP2) = 1 -
X
do(p)c(p)p- +P~2s - PXd0{p)c{p)p~:ia +p2~4s.
This process can be continued, leading to a complete evaluation of the P and Q
polynomials.
3.6. An application of the continuation of Z(s1w): quadratic twists
of GL(3). In this section we describe the consequences of the continuation to
C2
of
the multiple Dirichlet series Z{sJw) in more detail when TT is on GL(3). Recall that
if 7r' is a cuspidal autornorphic representation of GL(2) then the Gelbart-Jaequet
lift Ad (rr') is an autornorphic representation of GL(3) [35]. At good places v this
map is specified by the behavior of the local L-functions: if
L (
S )
) = ( ( l - a H - , ) ( l - A I » r ' ) ) " 1
then
L{s, A d 2 « ) ) = ((1 - avf3-'\v\-s)(l - \v\-*)(l - a ^ / ^ r ) ) " 1
(If TV' is self adjoint this is the symmetric square lift.) In [17j the following is proved:
THEOREM 3.3. Let ix1 be on GL2 (AQ). Let M be a finite set of places including
2, 00, primes dividing the conductor of TT'. Then there exist infinitely many quad-
ratic characiers Xd such that d falls in a given quadratic residue class mod v for all
v e M (mod Sifv = 2) and such that L(| , Ad2(7r') x Xd) ¥" ®-
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