12 2. PRELIMINARIES

The L-function L(s, χ) of χ, initially defined on Re(s) 1 by the convergent

Euler product of local L-factors L(s, χv) over all places v, is continued to a mero-

morphic function on C satisfying the functional equation

L(s, χ) = W

(χ)DF/2−sN(fχ)1/2−s 1

L(1 − s, ¯). χ (2.3)

The function L(s, χ) is holomorphic except possible simple poles at s = 0, 1

with

Ress=1L(s, χ) = δ(χ = 1) vol(F

×\A1),

Ress=0L(s, χ) = −δ(χ = 1)

DF/2vol(F 1 ×\A1).

(2.4)

For any oF -ideal a, let xa = (xa,v) ∈ Afin

×

be an idele such that xa,v =

ordva

v

for

all v ∈ Σfin, where ordv(a) ∈ Z is defined by aov = pv

ordv(a).

We designate the value

χ(xa) by ˜(a). χ

If χ is the trivial character 1, then L(s, 1) coincides with the Dedekind zeta

function multiplied with the gamma factor, which is denoted by ζF (s) in this article.

The Euler product of L(s, χv) over v ∈ Σfin is denoted by Lfin(s, χ).

2.4.3. Let Ξ0 be the set of all the unramified unitary idele-class character of

F

×.

There exists a constant C1 such that Lfin(s, χ) has no zero in a region of the

form Re(s) 1 − C1/ log{q(χ)(2 + |Ims|)}, |Ims| = 0 ([21, Theorem 5.10]). Then,

from the proof of [50, Theorem 3.11], we obtain the estimations

|Lfin(1 + it,

χ)−1|

= O(log q(χ| |A

it)),

Lfin(1 + it, χ)/Lfin(1 + it, χ) = O(log q(χ| |A

it)),

t ∈ R

(2.5)

with the implied constants independent of χ ∈ Ξ0. It is known that there exists

θ ∈ R such that the L-series Lfin(s, η) for an arbitrary idele-class character η of F

×

admits a bound on the critical line Re(s) = 1/2 of the form

|Lfin (1/2 + it, η)| q(η||A

it)1/4+θ,

t ∈ R (2.6)

with the implied constant independent of η. Actually, the bound with θ = 0 (the

convexity bound) is known for a while (see [33, 3.1]). Any bound (2.6) with θ 0

is called a subconvexity bound, which is established by Michel and Venkatesh in

this generality ([32]).

2.4.4. Let UF

+

be the set of totally positive unit of oF , viewed as a subset of

F∞ by the natural embeding. Then, the additive group log UF

+

= {(lv)v∈Σ∞ | lv ∈

R, (exp(lv))v ∈ UF

+

} is a lattice in the dF − 1 dimensional real vector space V =

{(xv)v∈Σ∞ | xv ∈ R,

∑

v

xv = 0 }. Let L0 be the dual lattice of log UF

+

in V , i.e.,

L0 = {(bv)v∈Σ∞ ∈ V |

v

bvlv ∈ Z (∀(lv) ∈ log UF

+)

}.

Let χ be a unitary idele-class character of F ×. The exponents a(χv) (v ∈ Σ∞) of

χ at the archimedean places are purely imaginary numbers with the constraint

v∈Σ∞

a(χv) = 0,

which comes from our convention χ|R+

×

= 1. If we set bv(χ) = Im{a(χv)} for

v ∈ Σ∞, then the vector b(χ) = (bv(χ)) belongs to the lattice L0. The mapping

χ → b(χ) is a surjection from Ξ0 onto L0 whose kernel Ξ0

0

coincides with the set of

finite order characters in Ξ0. Note that Ξ0

0

is a finite abelian group with order hF ,

the class number of F .