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 Vπ

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 Vπ

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)

.