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) ¥" ®-