20 2. PRELIMINARIES

where

Aχ,c(ν) =

˜2(c)N(c)−ν

χ

v∈S(c)

L(1 + ν, χv) 2

L(1 − ν, χv

−2)

.

Proof. By a well-known procedure found in the proof of [5, Theorem 3.7.1],

the proof is reduced to a computation of local intertwining operators on the vectors

f0,χv

(ν)

and

˜(ν)

f

1,χv

. Thus, we are done by Lemma 2.8.

Though ϕ = Eχ,c(ν; −) − Eχ,c(ν;

◦

−) is not left GF -invariant, it is still BF -

invariant; thus the integral

Z∗(s,

η; ϕ) makes sense.

Lemma 2.11. Let η be a unitary idele-class character satisfying (2.18), (2.19)

and (2.20). Then,

Z∗(s,

η; Eχ,c(ν) − Eχ,c(ν))

◦

(Re(s) 0) equals

(DF

N(c))−ν/2

˜(dF/Qc) χ G(η) Bχ,c(s,

η

ν)

L(s + ν/2,χη) L(s − ν/2,χ−1η)

L(ν + 1,χ2)

with

Bχ,c(s,

η

ν) =

DF−1/2 s

v∈S(c)

{(qv

−1

+ 1) L(1 + ν, χv)

2

ηv(

v)qv/2−s 1

− Q(Iv(χv||v

ν/2))

}.

(2.27)

Proof. Similar to Lemma 2.5.

Lemma 2.12. Let ν ∈ iR and η as in Lemma 2.11. The meromorphic function

Bχ,c

η

(

−z +

1

2

, ν

) L

(

−z +

ν+1

2

, χη

)

L

(

−z +

1−ν

2

, χ−1η

)

L(ν + 1,χ2)

in z ∈ C is holomorphic except possible simple poles at z = (ν ± 1)/2, (−ν ± 1)/2.

The relevant residues satisfy the relations

Resz=(ν+1)/2 =

−(−1)S(c)

N(c) Resz=(−ν−1)/2

= δχ,η vol(F

×\A1)

˜(c) χ (DF

N(c))(ν+1)/2,

Resz=(1−ν)/2 =

−(−1)S(c)

N(c) Resz=(−1+ν)/2

= δχ,η vol(F

× \A1)

˜(c) χ (DF

N(c))(ν+1)/2

DF

−1/2

Aχ,c(ν)

ζF (ν)

ζF (ν + 1)

.

Proof. The first statement is evident. The formulas of residues are obtained

by a direct computation with the aid of the easily confirmed relations

Bη,c

η

(

−ν

2

, ν

)

= (DF

−1N(c))(ν+1)/2

˜(c), η

Bη,c

η

(

1 −

ν

2

, ν

)

= (DF

−1N(c))(ν−1)/2

˜(c) η

(−1)S(c)

Aη,c(ν),

Bη,c

η

(

ν

2

, ν

)

= (DF

N(c))(ν−1)/2

N(c) ˜(c) η Aη,c(ν),

Bη,c

η

(

1 +

ν

2

, ν

)

= (DF

N(c))(ν+1)/2 (−1)S(c) N(c)−1

˜(c), η

combined with the functional equation of L(ν, 1) = ζF (ν).