MULTIPLE DIRICHLET SERIES AND AUTOMORPHIC FORMS
25
I I ( 1 - i\2-"s'"wr\i - ^ M 2 - " ) - \
where j
p
, 5P are the Satake parameters of the representation TTP. In fact, Brubaker,
Friedberg and Hoffstein show that Z*(s, w) has a meromorphic continuation to the
half plane 3?(s -f- w) 1/2 and is analytic in this region except for polar lines at
w = l,w = 0,w = b/3-2s,w = 3/2 - 2s, w = 4/3 - 2s, w = 7/6 - s, w = 1 - s,
w = 5/6 s. (With a little more work, they could establish continuation to
C2;
see
below.) They also show that the residue at w = 1 satisfies
Resw=1Z*(s, w) = cs Ls(Ss, TT, sym3) (SM (s(12s - 2)
and is an analytic function of s for 5te —1/2, except possibly at the points
s = 1/3,1/4,1/6,0, which require a more detailed analysis. The properties of the
symmetric cube L-series have been completely described by Kim and Shahidi.
4.2.2. The first two series and the first functional equation. This step is based
on the exact functional equation for the cubically-twisted L-series. Write d =
did^d3 as above. Ignoring bad primes such as those dividing the level of n and the
infinite place, L(s, 7r, \d ^2) has a functional equation of the form
Here IT denotes the contragredient of 7r, en (the central value of the usual epsilon-
factor for TT) has absolute value 1 and G(\d ) is the usual Gauss sum associated
to Xd ? normalized to have absolute value 1. The crucial factor \did2\1~2s arises as
part of the epsilon-factor of the twisted L-function since TT (&Xd ls ram ifi e d at the
primes dividing did2. This functional equation gives rise to a functional equation
for the double Dirichlet series Z\, reflecting Z\(s,w) into a second double Dirichlet
series
_ i s (
S
, t , x S
1
) G ( x S
1
) 2 P ( l -
S
; d
1
^ ) M
2
4 | ^
Z ^ w ) = 2l \d^4^
More precisely, the functional equation above induces a transformation relating
Zi(s, w) to ZQ(1 S, w+2s 1). (The exact transformation is somewhat complicated
due to bad primes.)
4.2.3. The second functional equation. Next we study the series ZQ(S,W) itself.
The appearance of G(xd ^ )
2
, the square of a cubic Gauss sum, introduces, via the
Hasse-Davenport relation, a conjugate 6-th order Gauss sum. However, the Fourier
coefficients of Eisenstein series on the 6-fold cover of GL(2) may be written as sums
of Gauss sums
G^(m,d)
d=l mod 3,(d,5)= l ' '
and accordingly series of this type possess a functional equation in w. One may
show, using this functional equation, that ZQ(S,W) possesses a functional equation
as (s,w) (s + 2w 1,1 w), transforming into itself.
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