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.