The Hasse-Weil 
-function
of an elliptic curve 
over 
is an Euler product formed
from its local reduction data. Let 
be the elliptic curve
conductor. At every prime 
of good reduction, let 
be the number of points on the reduction of 
over the finite field 
, and define 
. At a prime of bad reduction, set 
for split multiplicative reduction, 
for nonsplit multiplicative reduction, and 
for additive reduction. Then
 |
(1)
|
This is also called the 
-function or 
-series of 
, and it converges absolutely for 
.
The modularity theorem identifies 
with the 
-function
of a weight-two modular
newform whose modular form level is 
. It therefore has an analytic
continuation to the whole complex plane. Its completed form and functional
equation are
 |  |  |
(2)
|
 |  |  |
(3)
|
Here 
is the root number of 
. The central value 
and its order of vanishing are central to the Swinnerton-Dyer
conjecture.
: Wild 3-Adic Exercises." J. Amer. Math. Soc. 14,
843-939, 2001.