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.