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§3. Hasse–Weil L-Function and the Functional Equation 313

2
(1) f v (T ) = 1 − a v T + q v T , where a v = q v + 1 − N v ,if E has good reduction at
v.
(2) f v (T ) = 1 − e v T , where


−1If E has split multiplicative reduction at v,

e v = +1if E has nonsplit multiplicative reduction at v,

0if E has additive reduction at v.

Observe that in all cases

1
N v = q v f v .
q v
The complete non-Archimedean part of the L-function is v 1/L v (s), where L v (s) =
f v (q −s ).
v
(3.2) Definition. Let E/K be an elliptic curve over a number field K. The Hasse–
Weil L-function of E over K is
1
L E/K (s) = ,
L v (s)
v
where the local factor L v (s) is given in the previous section (3.1). Other notations
for L E/K (s) in current use are L E (s), L(s, E/K),and L(E/K, s).
The question of the local factors at infinity is taken up by J.-P. Serre in Fac-
teurs locaux des Fonctions Zˆ eta des Vari´ et´ es alg´ ebriques: S´ eminaire Delange-Pisot-
Poitou, 11 mai 1970. Serre in this 1970 DPP seminar gives a general definition for
any smooth projective variety over a global field.

(3.3) Definition. Let E/K be an elliptic curve over a number field K. The modified
Hasse–Weil L-function of E over K is

# E/K (s) = A s/2 K (s)L E/K (s),

where A = A E/K and K (s) are defined as follows:
(1) The constant A E/K is given by

A E/K = N K/Q ( f E/K ) · d 2 ,
K/Q
where d K/Q is the absolute discriminant of K and f E/K is the conductor of E
over K, see 14(2.4) and Serre, DPP, pp. 19–12.
(2) The gamma factor for the field K where n = [K : Q]is
n
−s
K (s) = [(2π) (s)] .