(1) First we have a functional equation under s 1 5, obtained from (3.3).
Because of the power of \d\ that is introduced we have, upon substituting
into (3.1), w - w + r(s - §). Thus
(3.5) Z(s,w) - Z ( l - s , w + r ( s - l / 2 ) ) .
(Strictly speaking we should write the right hand side as Z(\ s, w-\-r(s
1/2)) as 7r is replaced by its contragredient.)
(2) We also have a functional equation under the transformation w » 1 w,
obtained from (3.4). Applying this to (3.2) yields a transformation
(3.6) Z(s,w) - Z(s + w- 1/2,1 -w).
Note that each of these functional equations goes hand in hand with an exten-
sion of Z(s, w), originally defined by an absolutely convergent series in Sft(s), $l(w)
1, to a larger region. It is convenient to think of these transformations as operating
(repeatedly) on a region of definition to extend the function to a larger region, and
we will do so below, but strictly speaking one obtains first the continuation to the
larger region (by Phragmen-Lindelof), and then the functional equation on this
larger region.
Writing these functional equations carefully would require writing the archime-
dean factors and also describing a suitable scattering matrix; for the heuristic this
level of detail is not needed.
One can apply the functional equations (3.5) and (3.6) successively. They
generate a finite group of functional equations for GL(1), GL(2) and GL(3), i.e for
r = 1, 2, 3 but an infinite group for GL(4) (in fact an affine Weyl group) and higher.
This suggests that it should be possible to define a precise, non-heuristic, Z(s,w)
that continues to
for GL(1), GL(2) and GL(3) but that significant obstructions
may appear for GL(4) and higher.
To go farther, let us consider poles. We expect that there is a pole at w = 1,
arises in equation (3.2) when d = 1. This polar line is reflected by the
functional equations into a collection of polar lines that will be finite if r = 1, 2, 3 and
infinite if r 4 (see [16]). For any fixed SQ the possibility of a pole at s = so,
can be investigated by computing the sum of the contributions from the polar lines
that intersect (so, 1). If Z(s,w) does in fact have a pole at (50,1), then, by (3.1),
this implies the non-vanishing of L(SQ,7T X Xd) for infinitely many Xd- Similarly if
one can continue to (s,w) = (1/2,1) and if all epsilon factors at 1/2 are —1 then
one can differentiate with respect to s and set s = 1/2. There should still be a pole
at w = 1 provided that the different polar divisors do not cancel when s = 1/2. In
that case, one may then obtain a non-vanishing theorem for 1/(1/2, TT X Xd) from the
pole of -^Z(s,w) at s = 1/2, w 1. Standard methods involving contour integrals
can also give mean value theorems.
In the case of GL(4) and higher the group of functional equations is infinite.
If we take this infinite group and use it to translate the line w = 1, the poles
accumulate and create a barrier to continuation. See [16], Section 4, for some
elaboration of this point. Because of this we do not expect continuation to all of
when r 4. However, if we could get continuation up to the conjectured barrier,
that would be very significant; we would get a tremendous amount of information
(Lindelof in twisted aspect, simultaneous non-vanishing at the center of the critical
strip). At the moment this problem seems extremely challenging.
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