MULTIPLE DIRICHLET SERIES AND AUTOMORPHIC FORMS 15
Note that the estimate (3.11) holds for a(s,7r,d). Then we have
a(s, 7T, d) = ] P M e i) Xd(ei) 7r(eie|) |ei|~ s |e
2
| 1 " 2 s
(3.16) eie2e3=dl
= 7r(d?)|di|1-2- 5 ^ MeOXdCcijTr-^exeDleil'-^csl 2 - 1 .
eie2e3=di
Thus a(5,7r, d) satisfies the functional equation
(3.17) a(5,7r,d) = 7r(^)|di|
1
-
2 s
a(l-5,7T-
1
,d).
Since the conductor of \d is do (remember, we will ultimately avoid even places by
passing to a ring of 5-integers), L(S,TT X Xd) is equal to a factor involving the bad
places times
e{iiXd)\d{)\1^2~sL{\
5,
7i~1
X Xd)i where e^Xd) is the central value of
a global epsilon-factor.
Recall that Z(s,w) (or Z(s, w; 7r, £) to be more precise) is given by
Z(S,W;TT,€) = ^ L ( S , T T x Xd)a(s,ix, d)^(d)\d\~w.
d
Here £ is a second idele class character. Substituting in the functional equations for
L(s, Trxxd) and for a(s, 7r, d), one obtains a functional equation relating Z(s, w; n, £)
to Z(l - s,w + 5 - 1/2,TT,0 (cf. (3.9)). Notice that a factor of |d
0
| 1 / 2 _ s comes
from the functional equation for the GL(1) ^-function, arising since the conductor
changes by do upon twisting. This factor fits exactly with the |di|
1 _ 2 s
arising
from the functional equation (3.17) of the weight factor a(s,7r,d), and it is this
combination that shifts w to w+s —1/2. We also have that e{-KXd)^{d\) is essentially
constant—this is true for d congruent to 1 modulo a sufficiently large ideal, and
so the epsilon factors do not create a series of a fundamentally different type after
sending s 1— 1 s. See [31], Corollary 2.3, for more about the epsilon factors ([31]
works over a function field but the result is similar over a number field) and [31],
Theorem 2.6, for the exact functional equation.
We turn to the rewriting of Z(s,w) as a sum of Euler products in w, which
leads to the second functional equation (3.10). We always work in the domain in
which these sums converge absolutely ($l(s),$l(w) 1 will do). Substituting in
the definition of a(s, 7r, d) and expanding the L-function L(s, n x Xd) as a sum, we
obtain
Z ( S , W ; T T , 0 = Yl ] C Yl ^( d ) 7 r ( m )XdMM e i)Xd(ei)7r(eie|)
d=d0df rn e1e2\d1
x |m|-
s
|d| |e
1
|-
s
|e
2
|
1
-
2 s
.
The quadratic symbols give 0 unless (do,raei) = 1. Replace m by m' = me\. The
sum over m and e\ becomes Ylm'e 7 r ( m 0 Xd(^ / )l m/ | _s / i ( e i)5 where in the sum
ei|m;, ei|(di/e2). The sum over the Mobius function vanishes unless (m/,di/e2) =
1, in which case it is 1. So we obtain
E E E Z{d)el)Tt{m)xd{m')\d\-w\e2\l-2°\mr,
d=d0d\ e2|di m'
(m' ,dQd\ Ie2) = l
Now replace d by de?. This gives a sum over d, ra',e2 subject to the constraint
(d, ra') = 1. The sum over e2 gives L(2s + 2w
l,7r2£2).
Thus we obtain the
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