at the primes where p2 divides N). The Hasse–Weil zeta function of E then takes the form Here, ζ(s) is the usual Riemann zeta function and L(E, s) is called the L-function of E/Q, which takes the form where, for a given prime p, where, in the case of good reduction ap is p + 1 − (number of points of E mod p), and in the case of multiplicative reduction ap is ±1 depending on whether E has split (plus sign) or non-split (minus sign) multiplicative reduction at p. A multiplicative reduction of curve E by prime p is said to be split if -c6 is a square in finite field with p elements. There is a useful relation not using conductor: 1. If p doesn't divide (where is a discriminant of an elliptic curve) then E has good reduction at p. 2. If p divides but not then E has multiplicative bad reduction at p. 3. If p divides both and then E has additive bad reduction at p. See also Arithmetic zeta function References Bibliography J.-P. Serre, Facteurs locaux des fonctions zêta des variétés algébriques (définitions et conjectures), 1969/1970, Sém. Delange–Pisot–Poitou, exposé 19 Zeta and L-functions Algebraic geometry