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 fχ 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).