Page 340 -

§4. Classical Abelian L-Functions and Their Functional Equations 315

§4. Classical Abelian L-Functions and Their Functional
Equations

The classical zeta and L-functions were defined and studied by Riemann, Dirichlet,
Dedekind, and Hecke. The Dedekind zeta function of an algebraic number field is
the generalization of the Riemann zeta function for the rational numbers, and Hecke
L-functions for a number field are generalizations of the Dirichlet L-functions of
characters on the rational integers. We give the definition of each of these functions as
a Dirichlet series, give the Euler product expansion, and state the functional equation.
Of special interest is the Hecke L-function with “Grossencharakter” for it is this L-
function which is related to the Hasse–Weil L-function of an elliptic curve with
complex multiplication.
(4.1) Riemann Zeta Function. The Dirichlet series expansion of the Riemann zeta
function is for Re(s)> 1

1
ζ(s) = .
n s
1≤n
The Euler product, which is the analytic statement that natural numbers have a unique
factorization into primes, is given by

1
ζ(s) = for Re(s)> 1.
1 − p −s
p
The Riemann zeta function has a meromorphic continuation to the entire complex
plane, and it satisfies a functional equation which is most easily described by in-
troducing the function ξ(s) = π −s/2 (s/2)ζ(s) related to the zeta function. The
functional equation for the zeta function is simply

ξ(s) = ξ(1 − s).
One proof of the functional equation results from using the elementary theta function
2
−πn t
θ(t) = e and the integral for the -function . Then
n
∞ dx 1 1 ∞
dx

ξ(s) = x x/2 θ(x) = + + x (1−s)/2 + x s/2 θ(x) .
0 x s 1 − s 1 x
This follows by dividing the first integral into two integrals at x = 1. The above
√
representation holds because the theta function satisfies θ(1/t) = tθ(t), which is a
modular type relation. The subject of modular forms was considered in Chapter 11.
The theta-function relation is proved by applying the Poisson summation formula to
e −πtx 2 and using the fact that e −πx 2 is its own Fourier transform.
In the rational integers Z, each nonzero ideal a is principal a = (n) where n ≥ 1,
the absolute norm Na = n,and a is a prime ideal if and only if a = (p) for a prime
number p. These remarks lead to Dedekind’s generalization of ζ(s) to a number field
K using the ideals a in O, the ring of algebraic integers in K, and the absolute norm
Na = #(O/a), the cardinality of the finite ring O/a.