2.4. L-FUNCTIONS FOR IDELE-CLASS CHARACTERS 11
2.4.1. Let v Σfin. For a unitary character χv of Fv
×,
let f(χv) N be the
minimal integer f N such that χv|Uv(f) is trivial, where Uv(f) = 1 + pv
f
if f 0
and Uv(0) = ov
×.
Thus
pv(χv) f
is the conductor of χv. The root number of χv is
defined by
W (χv) = qv
−f(χv)/2
ξ∈ov
×/Uv
(f(χv))
¯v(ξv χ
−dv−f(χv))
ψF,v(ξv
−dv−f(χv)),
which is a complex number of modulus 1 independent of the choice of the uni-
formizer
v
. The integral
ov
×
χv(uv
−dv−f(χv))
ψF,v(uv
−dv−f(χv))
d×u
is called
the Gauss sum associated with χv and is denoted by G(χv). We have
G(χv) = qv
f(χv)/2
vol(Uv(f(χv));
d×xv)
W (¯v) χ (2.2)
and W (χv) = G(¯v)/|G(χv)|. χ The local L-factor of χv is defined by
L(s, χv) =
(1 χv(
v
) qv −s)−1, (f(χv) = 0),
1, (f(χv) 0).
When χv is unramified then it has of the form
χv(x) =
|x|vva
with
av [0, 2πi(log
qv)−1);
in this case, we set a(χv) = av, calling it the exponent of χv.
Let v Σ∞ and χv a unitary character of Fv
×

=
R×.
If we write χv(x) =
|x|vv
a
sgn(x)
v
, x R with av iR and
v
{0, 1}, then the local L-factor and the
root number of χv is defined as
L(s, χv) = ΓR(s + av +
v
), W (χv) = i
v
,
respectively. We set a(χv) = av (resp. (χv) =
v
), calling it the exponent (resp.
the sign) of χv.
2.4.2. Let χ be a unitary idele-class character of F ×; in this article throughout,
whenever we consider such a character, it is assumed that it has a trivial restriction
to R+.
×
The conductor of χ is the oF -ideal such that fχov =
pv(χv) f
for all
v
Σfin. For latter purpose, following [21, Chapter 5], it is convenient to introduce
the analytic conductor q(χ) of χ by setting
q(χ) = N(fχ)
v∈Σ∞
(3 + |a(χv)|)
with (a(χv))v∈Σ∞ the exponents of χ at archimedean places. Note that W (χv) =
G(χv) = 1 for almost all v Σfin. Set
W (χ) =
v∈Σfin∪Σ∞
W (χv), G(χ) =
v∈Σfin
G(χv).
Lemma 2.1. Set =

v∈Σ∞
v. Then,
W (χ) = i
DF/2 1
N(fχ)−1/2
{ (oF
/fχ)×}
G(¯). χ
Proof. The local volume vol(Uv(f);
d×t)
equals qv
−dv/2
or qv
−dv/2−f
(1−qv
−1)−1
according to f = 0 or f 0, respectively. Since (oF
/fχ)×
=
f(χv) 0
qv
f(χv)(1

qv
−1),
we have the result from (2.2).
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