16 GAUTAM CHINTA, SOLOMON FRIEDBERG, AND JEFFREY HOFFSTEIN

equation (dropping the prime on the variable m')

(3.18) Z(s,w-n,0 = L{2s + 2w-l)K2(,2) ]jT £(d) 7r(m)

Xd

(m) \d\-w\m\~s.

(d,m)=l

Modulo dealing carefully with quadratic reciprocity, we see that we have a func-

tional equation Z(s,w;7T, p) = Z(w,s; p, 7r). (For the precise statement, see [31],

Theorem 3.3.) This gives (3.8) and the second desired functional equation (3.10),

and allows us to establish the continuation of Z(s^w) to C2.

We remark that a similar proof applies to n-fold twists, provided that one writes

d = dodi with d\ n-th power free and one uses the weight function

a(s,n,d)= ] T

^e1)Xd(ei)7r(e1e^\e1\-s\e2\n-1-ns.

e1e2\d1

See [34], Proposition 2.1, as well as Section 4.1 below.

3.4.2. Sums of GL(2) quadratic twists. In this section we follow the approach

of Fisher and Friedberg [32] to present the GL(2) computation. Suppose now that

7 T is cuspidal on GL(2) with L(S,TT) =

]JV((1-TT1(V)\V\-S)(1-7T2(V)\V\-S))-1.

Here

7Ti(^), 7T2(v) are the Satake parameters for nv. (Once again we are really taking

the L-function with the primes in a finite set S of bad places removed, but we omit

this from the notation.) Extend 7TI,7T2 multiplicatively to be functions defined on

ideals prime to S. Let

(3.19) A(s, 7r, d) — a(s, 7Ti, d) a(s, 7T2, d)

where the factors on the right hand side are given by (3.15). It will turn out that

A(s17r, d) is closely related to the desired GL(2) weight function a(s, 7r, d); see (3.23)

below. For £ on GL(1), we set

ZA(s, w; 7T, 0 = J^

L

(

5

' *

x

*d) A(s, TT, d) £(d)

\d\~w.

From the functional equation (3.17) for the GL(1) weight function, we obtain

(3.20) A(s,7r,d) = X T T ( ^ ) M I |

2

~

4 S

^ (

1

- «,^d) ,

where as above d — d$d\ with do square-free, and where \n is the central charac-

ter of 7r. From this and the functional equation for the L-function L(S,IT X Xd)i

one immediately obtains a functional equation for ZA(S,W) with respect to the

transformation (s,w) ^^ (1 — s, w + 2s — 1).

A second functional equation is obtained by proving an analogue of (3.18).

Namely, we have the key (and remarkable) formula

(3.21) L(2s + 2w-l,

xA2)

ZA(s, w; TT, 0 = L(4s + 2w-2, xl?)

x X ]

7ri(TOi)7r2(™2)L(w,£xmim2)a(w,£,TOiTO2)|m1m2rs.

mi ,7Tl2

Here a(w,£,mim2) is the GL(1) weight factor, given by (3.15). Though the full

details are too long to include here (see [32], Section 2), we will present a sketch of

the proof of this result.