10 GAUTAM CHINTA, SOLOMON FRIEDBERG, AND JEFFREY HOFFSTEIN

3. Quadratic Twists

3.1. A heuristic. In Section 2.3 we have seen that a number of double Dirich-

let series arose via Rankin-Selberg integrals. Such series necessarily have continua-

tion coming from the integral. Could this have been predicted without the integral?

And what happens if one cannot find such an integral?

In 1996, Bump, Friedberg and Hoffstein [16] presented a heuristic that explains

what to expect in the quadratic twist case. We describe it now. Consider a GL(r)

L-function

L(s,7r) = ]Te(ra)|ra|-

s

.

n

The family of objects of interest is L(s,7r X \d), where Xd varies over quadratic

twists; we write

L(S,TT x Xd) = y2c{m) ( —

m x

Note that (^) is zero if (d,m) 1, so this equation is not exactly correct if d is

not square-free, but we will not keep track of this complication at the moment. Set

/Q1N ry( X TT^ L(S,7T XXd)

(3.1) Z(s,w) =

M*

In fact, this is not the actual definition of the correct multiple Dirichlet series as

we are ignoring weight factors and also not specifying the m that we are summing

over. We are now in the land of the heuristic and things will get even looser. If

we temporarily pretend that all integers are square-free and relatively prime, then

we can expand the L-series in the numerator of Z(s,w) and write (for 9ft(s), 5ft(u)

sufficiently large)

z(S,w) =

YH2c^(^)m~Sd'w-

dm ^ '

In this heuristical universe we may as well assume that reciprocity works perfectly

with no bad primes. Assuming this, we can reverse the order of summation, ob-

taining

(3.2) Z(s,w)=

"}Tc(m)L(w,Xm)m-s-

m

Note that we started with a sum of L(s, n x Xd), that is, a sum of twisted GL(r)

//-functions, in (3.1), and ended with a sum of L(w,Xm)i that is, a sum of twisted

GL(1) L-functions in (3.2)! That is, our sum of Euler products in s is at the same

time a sum of Euler products in w\ Again, this is only a heuristic, as it assumes

( T ) = (m) anc^ a ^ n u m bers are square-free and relatively prime. However it turns

out that reality can be made to fit this heuristic remarkably well.

We will now explore the functional equations of these twisted //-functions. For

d square-free there is a functional equation sending

(3.3) L(5,7TXXd) - | d r

(

* -

s )

L ( l - 5 , 7 r x

X

d ) ,

as well as one sending

(3.4) L(w,Xm) -• \m\^-sL(l-w,Xm).

Thus Z(s,w) satisfies two types of functional equations: