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(s,7r) = ]Te(ra)|ra|-
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) =
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) =
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-
(3.2) Z(s,w)=
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
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:
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