18 GAUTAM CHINTA, SOLOMON FRIEDBERG, AND JEFFREY HOFFSTEIN
Summing over e, we see that
Z(s, w; TT, 0 - L(2s + 2w-l,
X^2)
Yl
L
(
s
'
n x
Xd) a(s, TTU d) a(s, TT2, d)£(d)
\d\~w
d
= L(2s + 2w - 1,
X^2)
ZA(s, w;
TT,
0-
We may hence apply equation (3.21) in order to see that Z(S,W;TT,£) is equal to a
sum of GL(1) L-functions in w, as desired.
3.5. More on the interchange of summation: an example of the
uniqueness principle. The interchanges of summation exhibited in the previ-
ous section raise the following questions: (a) are the weight factors given there
canonical? and (b) how can one find such factors, if one does not know them in
advance? In this section we answer these questions when TT is on GL(2). We will
explain how to determine the weight factors of the multiple Dirichlet series directly,
thereby establishing a uniqueness principle. More precisely, we will suppose that
the weight factor has three properties: (i) it has an Euler product; (ii) it gives the
proper functional equation for the product L(s, n x \d) a(s, 7r, d) even when d is
not square-free; and (iii) it has the correct properties with respect to interchange
of summation. Under these assumptions, we will show that the weight factor for
generic primes is unique, and in fact may be determined completely. (We will still
ignore bad primes, for convenience.) The approach given here works for GL(1)
(an easy exercise), and it also generalizes to other situations, such as GL(3) ([17]),
where the weight factors are too complicated to guess.
So suppose that 7r is an automorphic representation of GL(2), with standard
L- function
c(m)
L(s,ir) = £ :
For convenience we take the central character of n to be trivial.
Write d = d$d\ with do square-free. We begin by assuming that
a(s, 7r, d) = P(s, d$d\),
where P{s,d$d\) is a Dirichlet polynomial, that is a polynomial in m~s for a finite
number of m (the factors P{s1dod\) depend on 7r, but we suppress this from the
notation).
What properties should P(s, dod\) have? For the functional equation to work
out correctly we require
(3.24) P(s,d0dj) = dl-4sP(l-s,d0dl).
We also require that there be an Euler product expansion for P, namely
(3.25) P(s,d0dj)
= H (l + a(doP2a,l)p-s +a(doP2a,2)p-2s + ... + a(d
0
p 2 a ,4a)p- 4 ^) ,
Pa\\dx
where the a's are coefficients to be determined. Note that each factor is forced to
end at
p~4as
by (3.24). In fact (3.24) implies the recursion relation
a(d0p2a, k) = pk-2aa(d0p2aAa ~ k)
for 0 k 4a.
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