2L-functions
IV Topics in Number Theory
2 L-functions
Recall that Dirichlet characters
χ
: (
Z/NZ
)
×
→ C
×
give rise to Dirichlet
L
-
functions. Explicitly, we extend
χ
to a function on
Z
by setting
χ
(
a
) = 0 if
[a] 6∈ (Z/NZ)
×
, and then the Dirichlet L-function is defined by
L(χ, s) =
∞
X
n=1
χ(n)
n
s
=
Y
p
(1 − χ(p)p
−s
)
−1
.
As one can see from the
n
and
p
appearing, Dirichlet
L
-functions are “about
Q
”, and we would like to have a version of
L
-functions that are for arbitrary
number fields
K
. If
χ
= 1, we already know what this should be, namely the
ζ-function
ζ
K
(s) =
X
aCO
K
1
N(a)
s
=
Y
pCO
K
1
1 − N(p)
−s
,
where the first sum is over all ideals of
O
K
, and the second product is over all
prime ideals of O
K
.
In general, the replacement of Dirichlet characters is a Hecke character.