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

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.
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