Although the Riemann hypothesis concerning the zeros of L(%, s) has been
proved by Weil in the function field case, there seems to be no breakthrough in
sight in the number field case. The conjunction of the Artin conjecture for all V
and the Riemann hypothesis for all % is equivalent to the positivity of a certain
distribution on WF (cf. [W3]).
(3.6) Comparison of different conventions for local constants. The modern ref-
erences for the material we have been discussing are Deligne [D3] and Langlands
[L], and we have here followed the conventions of [D3]. Happily, the definition of
L-functions, both local and global, in [D3] coincides with that in [L]. But Deligne's
local constants e(V, $, dx), which we will designate in this section by eD instead of
just £, differ somewhat from Langlands' e(V, $) which we will denote by eL here.
The relationship is
(3.6.1) eL(V, jj) = eD{Va)i/2, /, dx^),
where dx^ is the additive measure which is self-dual with respect to j). The other
way around we have
(3.6.2) eD(V, 0, dx) = (dx/dx^ eL{Va-l/2, #
In the nonarchimedean case the constant dxjdx^ is given explicitly by
q-n(f)/2 j0dx. Also, in that case if V corresponds to a quasi-character % of F* we
(3.6.3 ) 4 # = lwKn77W -
where c is an element of F* of valuation a{y) + n(fi) as in (3.2.6).
Langlands puts
(3.6.4) eL(s, V, 0) = eL(Va)s-a/2) I) = £ D ( ^ & dx^).
Then the "constant" e(V, s) in the global functional equation (3.5.4) is given by
e(V, s) =
Vv fv)forany nontrivial character J of A/F, because if dxv is
self-dual on Fv with respect to Jjv for each place v, then dx = \[v dxv is self-dual
on A with respect to $, and is therefore the Tamagawa measure on A.
The behavior of eL under twisting by an unramified quasi-character is given by
(3.6.5) eL(Vas, 0) = eL(V, 0)/f K)"*
as in (3.4.5), but its dependence on p is according to
(3.6.6) eL(V, 0 J = (det V)(a)e(V, 0),
instead of as in (3.4.4). If F* is the contragredient of V, then
(3.6.7) eL(V9 0)eL(F*, 0-i) = 1.
Hence, by (3.6.6)
(3.6.8) eL(V, 0) eL(V\ /,) = (det F ) ( - 1)
and on the other hand,
(3.6.9) | sL( V, (])) | = 1, if V is unitary.
The £L-system has the advantage that it avoids carrying along the measure dx9
but it has the following disadvantage: in the nonarchimedean case, if a is a dis-
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