2.6. CUSPIDAL AUTOMORPHIC REPRESENTATIONS AND THEIR L-FUNCTIONS 17
Proof. By a standard argument found in [5, pp.335–336], the integral
Z∗(s,
η; ϕπ,c)
is decomposed to a product of local ones
Zv
∗(s,
ηv; φv) = ηv(xη,v)

Zv
(
s, ηv; πv
(
1 xη,v
0 1
)
φv
)
over all v, where φv is φ1,v or φ0,v according as v belongs to S(c) or not. If ηv
is unramified, then Zv ∗(s, ηv; φ0,v) = Zv(s, ηv; φ0,v) is given by (2.8). If πv and
ηv are both unramified, then Zv ∗(s, ηv; φ1,v) is evaluated in Lemma 2.2. It re-
mains to calculate the integral Zv
∗(s,
ηv; φ0,v) when πv is unramified and f(ηv) 0.
Since φ0,v [ t 0
0 1
] 1
−f(ηv)
v
0 1
= ψF,v(tv
−f(ηv))
φ0,v ([ t 0
0 1
]), by substituting (2.11),
we have
Zv
∗(s,
ηv; φ0,v) =
Fv×
φ0,v ([ t 0
0 1
]) ψF,v(tv
−f(ηv))
ηv(tv
−f(ηv))|t|v−1/2 s d×t
=
m −dv
qv
−(m+dv)/2
qvv(m+dv+1) ν
qv
−νv(m+dv+1)
qvv
ν
qv
−νv
ηv(
v
)mqv −(s−1/2)m
× {
ov×
ψF,v(uv
m−f(ηv))
ηv(uv
−f(ηv)) d×u}.
The integral in the last line is evaluated as δ(m = −dv) ηv( dv
v
) G(ηv). Thus,
only the term for m = −dv survives in the last summation, yielding the identity
Zv
∗(s,
ηv; φ0,v) = qvv
d (s−1/2)
G(ηv). Since πv is unramified and ηv is ramified, the
L-factor L(s, πv ηv) = 1. This completes the proof.
Lemma 2.6. Let π be an irreducible cuspidal automorphic representation with
trivial central character such that
K0(n)K∞
= {0}. Let η be an idele-class character
of F × satisfying the conditions (2.18), (2.19) and (2.20). Then, the number
Pη(π;
K0(n)) =
ϕ∈B
Z∗(1/2, 1; ϕ)
Z∗(1/2,η;
ϕ)
is independent of the choice of an orthonormal basis B of
K0(n)K∞
. We have
Pη(π;
K0(n)) = DF
−1/2
G(η) {
v∈S(nfπ
−1)
1 + ηv( v)
1 + Q(πv)
}
L(1/2,π) L(1/2,π η)
ϕπ new 2
.
(2.21)
The number
G(η)−1 Pη(π;
K0(n)) is non negative.
Proof. The first assertion is obvious. To show the second statement, we take
B = {ϕπ,c/ ϕπ,c | nfπ −1 c }. Then, by Lemmas 2.4 and 2.5 ,
Pη(π;
K0(n)) =
nfπ
−1
⊂c
ϕπ,c
−2Z∗(1/2,
1; ϕπ,c)
Z∗(1/2,
η; ϕπ,c)
= {
nfπ
−1
⊂c
v∈S(c)
(ηv( v) Q(πv))(1 Q(πv))
1 Q(πv)2
} G(1)G(η)
L(1/2, π) L(1/2, π η)
ϕπ new 2
.
In the last line, the sum in the bracket is evaluated as
v∈S(nfπ
−1)
1 +
ηv(
v
) Q(πv)
1 + Q(πv)
=
v∈S(nfπ
−1)
1 + ηv(
v
)
1 + Q(πv)
.
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