16 2. PRELIMINARIES
Proof. From assumptions, we have
(1/2,πv,ψF,v) v(1/2,πv ηv,ψF,v) =

⎪ηv(−1),



(v Σ∞ S(fη)),
ηv(
v
), (v S(fπ)),
1, (v Σfin S(fπ fη).
Thus, (1/2,π) (1/2,π η) = ˜(fπ) η = ±1. Since η2 = 1, the central character of
π η is also trivial. Hence the claim is a consequence of the functional equation
relating the values at s and 1 s of L(s, π) L(s, π η).
2.6.2. Fix an oF -ideal n such that n is square free. Then, the product K0(n) =
v∈Σfin
K0(nov), is an open compact subgroup of Kfin. Suppose
K0(n)K∞
= {0}
from now on. Then, Vπv
K0(nov)
= {0} for all v Σfin. This implies that divides n.
For any ideal c dividing nfπ −1, let ϕπ,c be the function in corresponding to the
decomposable tensor
{
v∈S(c)
φ1,v} {
v∈S(c)
φ0,v}
under the identification

=
v
Vπv . The function ϕπ,oF is called the newform of
π and is denoted by ϕπ new.
Lemma 2.4. The functions ϕπ,c (nfπ
−1
c) form an orthogonal basis of the
invariant part
K0(n)K∞
, and
ϕπ,c 2
ϕπ new 2
=
v∈S(c)
{1
Q(πv)2}.
(2.17)
Proof. Since π is irreducible, there exists a positive constant C such that the
L2-inner
product on coincides with C · {
v
( | )v} by the isomorphism

=
v
Vπv . Since c is relatively prime to fπ,
ϕπ,c
2
= C
v∈S(c)
{1
Q(πv)2}
by Lemma 2.2. Taking c = oF , we obtain C = ϕπ,oF
2.
This shows (2.17).
Let η be a unitary idele-class character of F
×
such that
η2
= 1, (2.18)
ηv(−1) = 1 for any v Σ∞, (2.19)
is relatively prime to n and ˜(n) η = 1. (2.20)
Instead of the zeta integral (2.16), which vanishes identically for ϕ
K0(n)K∞
unless η = 1, we consider
Z∗(s,
η; ϕ) = η(xη)

Z
(
s, η; π
(
1
0 1
)
ϕ
)
, ϕ
K0(n)K∞
,
where
(resp. xη) is the adele (resp. idele) whose v-component is
−f(ηv)
v
or
0 (resp. 1) according to v Σfin or v Σ∞, respectively. The factor η(xη) is
included to make the expression independent of the choices of uniformizers v.
Lemma 2.5. Let η be as above. Then, for any c dividing nfπ −1, we have
Z∗(s,
η; ϕπ,c) =
DF−1/2 s
G(η) {
v∈S(c)
ηv(
v
)qv/2−s 1
Q(πv) } L(s, π η).
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