MULTIPLE DIRICHLET SERIES AND AUTOMORPHIC FORMS 17
First, substituting in the Dirichlet series for L(s, ir x Xd) and changing variables
to sum the two Mobius functions, we find that
ZA(s,w;7T^) =
Y] 7ri(mi) 7r2(m2) £(d)
Xd(miei~2) XdO^eJ2)
|eie2|
\m1m2\~s\d\~w
mi ,rn,2,d,ei,e2
where the summation variables are subject to the restrictions (m^, d) = e2, i 1, 2
([32]), Proposition 2.2). Introducing a variable e = (ei,e2), one can rewrite the
sum and pull out an L-function L(4s + 2w 2,x2£2)- Then replacing d by de^e2,
one arrives at a sum over variables rai, ra2, d, ei, e2 subject to the constraints e2\rrii
(i = 1,2), (ei,m2) = (e2,mi) = 1, and (d, raira2ef
2e^2)
= 1. Replacing this last
equation in the standard way by a sum of Mobius functions, one can once again
obtain an L-function L(w^Xm1m2)- Then multiplying by L(2s + 2w
1,XTT£2)
writing this last as a sum (over g) and changing several summation variables, we
obtain
(3.22) L(2s + 2w - 1, x ^ 2 ) ZA(s, w; TT, f) = L(4s + 2w - 2, x2£2) x
] T TTl(^l) 7T2(m2) L(w, £Xmim2) M0 (d)^(de^e^2)
mi .m2,d,ei ,e2,g
|m
1
m
2
|- s |d|- u ' |eie27r-2w
with summation conditions
#e2|ra;
(i = 1,2),
(deie2#)2|raira2,
(ei,
ra^-1)
=
(e
2
,raig
- 1
) = (d, (raira2y) = 1, where the prime denotes the square-free part.
But given rai,ra2, there is a one-to-one correspondence between triples (ei,e2,g)
such that
ge2|ra;
(z = 1,2), (ei,
ra2g~1)
(e2,
rnig-1)
1 and numbers / such
that
f2\m\m2\
the correspondence takes (e\e2,g) to / = eie2# (see [32], Lemma
2.5). Applying this, equation (3.22) can be rewritten
L(2s + 2w- l , x ^
2
) ZA(s,w;ir^) = L(4s + 2n; - 2,
X
2
£
2
)x
^2 Triirm) 7r2(rn2) L(w,£x Xmirri2
(dK(d/2)
rai .rri2,d,f
\rnim2\-s\d\-w\f\l-2w
where in the sum
dl f2\rn\m2,
(d, (raira2)') = 1. The sum over d and / gives the
GL(1) weight factor a(w,^,mim
2
), and equation (3.21) follows.
Finally, let us give the GL(2) weight factors and explain the relation between
formula (3.21) and the equality of (3.7) and (3.8) for suitable weight factors. The
GL(2) weight factor is given by:
(3.23) a(s, 7r,d) = Yl M 1 " 2 * * ^ ) A{s, TT, de~2).
e2\d
Since the quantity |e|
1 _ 2 s
A(s, 7r,
de~2)
satisfies precisely the same functional equa-
tion (3.20) as A(s, TT, d) itself, we see that Z(s, w; 7r, £) satisfies a functional equation
with respect to the transformation (s, w) K^ (1 s, it; + 2s 1). As for the equal-
ity of (3.7) and (3.8) (for suitable 6), substituting (3.19), (3.23) in to (3.7), and
interchanging summation one obtains
Z(s, w;K,0 = Y1 L^?r x X") *(*,*i, * ) *(*,"2,d)£(de2) X.(e) lei1"28-2™ |d|"w.
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