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316 16. L-Function of an Elliptic Curve and Its Analytic Continuation

(4.2) Dedekind Zeta Function. The Dirichlet series expansion of the Dedekind
zeta function for the number field K is
1

ζ K (s) = for Re(s)> 1.
Na s
a =0
If the degree [K : Q] = n, then there are at most n ideals a with Na equal to a given
prime p, and this is the reason why the Dirichlet series for ζ K (s) has the same half
plane of convergence os the Riemann zeta function. The Euler product, which is the
analytic statement that ideals in the ring of integers in a number field have unique
factorization into prime ideals, is given by

1
ζ K (s) = for Re(s)> 1.
Np s
p
The Dedekind zeta function has a meromorphic continuation to the entire complex
plane, and it satisfies a functional equation. For the functional equation we need
d = d K the absolute discriminant of K and the decomposition of the degree [K :
Q] = n = r 1 + 2r 2 , where r 1 is the number of real places of K and r 2 is the number
of conjugate pairs of complex places of K. In 1917, Hecke proved for
√ s
|d|
s r 1
r 2
ξ K (s) =

(s) ξ K (s)
r
2 2 π n/2 2
the functional equation
ξ K (s) = ξ K (1 − s).
The zeta functions contain information about the distribution of prime numbers
and prime ideals. In an L-function this information is further organized by weight-
ing the primes via character values. On a finite abelian group, a character is just a
∗
∗
homomorphism into C .If χ is a character of the finite abelian group (Z/mZ) of
units in the ring Z/mZ, then we extend the complex valued function χ to be zero on
∗
Z/mZ outside of (Z/mZ) , and we compose it with the quotient map Z → Z/mZ

to define a function on Z also denoted χ.On Z we have χ(nn ) = χ(n)χ(n ) and
χ(n) = 0 if and only if the greatest common divisor (n, m)> 1.
(4.3) Dirichlet L-Function. The Dirichlet series expansion of the Dirichlet L-func-
tion associated with a character χ mod m is
χ(n)
L(s,χ) = .
n s
1≤n
The Euler product is given by
1

L(s,χ) = −s
1 − χ(p)p
p
and both the Dirichlet series and the Euler product converge absolutely for Re(s)>
1. For the trivial character χ = 1, L(s, 1) = ζ(s) the Riemann zeta function.