NUMBER THEORETIC BACKGROUND
13
L(X-»US)
= rR(s)d=\-^ r(S/2).
(3.2.1) F « C. For z an embedding of Fin C and N ^ 0,
defn
(3.1.3) Fnonarchimedean. For 7r a uniformizer in F,
(1 x(7t))~l, if x ls unramified,
if / is ramified.
In every case, L is a meromorphic function of/, i.e.,
L(XDS)
is meromorphic in s,
and Z, has no zeros.
(3.2) Local abelian ^functions. We will denote by dx a Haar measure on F,
by d*x a Haar measure on F* (e.g., d*x =
||x||_1
dx) and by ^ a nontrivial additive
character of F.
Given cp and *£c, one has a "Fourier transform"
/0) = $Ax)4(xy)dx.
The local functional equation
( } lXf»irl) -^'V'*** L(X)
defines a number e(x, fi, dx)e C* which is independent of/, for/'s such that the two
sides make sense. If / i s continuous such that/(x) and/(x) are 0{e~m) as
||JC||

oo, then the two sides make sense naively for x s u c n that x(*) = II*II" with 0
a 1, and each side is a meromorphic function of x O
n e
takes the same multi-
plicative Haar measure d*x on each side. The dependence of e on (J and dx comes
from the dependence of the Fourier transform f(x) on $ and dx. One finds
(3.2.2) e(x /, rdx) = re(x & dx), for r 0,
(3.2.3) e(x, 0(ax),rfx)= z(*)|| a||-i eft, 0, rfx) for a e F*.
Easy computations carried out in [Tl] and [W2] show that the function e is given
by (3.2.2), (3.2.3) and the following explicit formulas:
(3.2.4) F ^ R. Let x be the embedding of F in C and N = 0 or 1. For (p =
exp(2^r/x) and £c the usual measure, e(x~Ncos, (/, dx) = iN.
(3.2.5) F & C. Let z be an embedding of F in C and JV ^ 0. For 9^ =
exp(2;n Trc/Rz) and dx = idz A dz (= Ida db for z = a + bi), e(z~Ncos, /), dx)
= i* .
(3.2.6) F nonarchimedean. Let 0 be the ring of integers in F. Put
w(^) = the largest integer n such that
(jj{7t~n0)
= 1,
tf(^) = the (exponent of the) conductor of x (= 0 if/ is unramified, the smallest
integer m such that
z
is trivial on units = 1 (mod
izm)
if % is ramified),
c an element of F* of valuation «(^) 4- #(/). If x is unramified,
(3-2-6.1) e(x,p,dx)=X^jedx.
(In particular, s(z, ^, dx) = l,ifjC)dx = 1, and n(p) = 0 when
z
is unramified.)
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