2.8. ADJOINT SQUARE L-FUNCTIONS 23

Using this, we calculate the integrals to obtain

Zv

(1)(s)

= vol(K0(pv)) vol(ov

×)

qvv

d (s−1)

(1 − qv

−s+2νv )−1(1

− qv

−s−2νv )−1(1

− qv

−s)−1

× {(1 + qv

−s

)(1 + qv

−s+1

) − 2qv

−s+1/2 (qv/2 1

+ qv

−1/2

)},

Zv

(2)(ξ;

s) = vol(K0(pv)) vol(ov

×)

qvv

d (s−1)

(1 − qv

−s+2νv )−1(1

− qv

−s−2νv )−1(1

− qv

−s)−1

× {(1 +

qv−1)(1 s

+ qv

−s)

− 2qv

−1/2(qv/2 1

+ qv

−1/2)}.

Then, a further computation reveals that Zv(s) is given by the formula (2.30)

multiplied with vol(K0(pv)) qv.s

Corollary 2.15.

ϕπ

new 2

= 2 N(fπ) [Kfin :

K0(fπ)]−1

L(1,π;Ad).

Proof. This is proved by taking the residue of (2.29) at s = 1. Use Lemma 2.13

to compute the residue of E1,oF (2s − 1; g).

Corollary 2.16. Let n be an square free ideal of oF . Let π be an irre-

ducible 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,

Pη(π;

K0(n)) = DF

−1/2

G(η)

[Kfin : K0(fπ)]

2N(fπ)

{

v∈S(nfπ

−1

)

1 + ηv( v)

1 + Q(πv)

}

L(1/2,π) L(1/2,π ⊗ η)

L(1, π; Ad)

.

Proof. This follows from (2.21) and Corollary 2.15.