# Analytic continuation of Dirichlet L-functions #

We show that if `χ`

is a Dirichlet character `ZMod N → ℂ`

, for a positive integer `N`

, then the
L-series of `χ`

has analytic continuation (away from a pole at `s = 1`

if `χ`

is trivial).

All definitions and theorems are in the `DirichletCharacter`

namespace.

## Main definitions #

`LFunction χ s`

: the L-function, defined as a linear combination of Hurwitz zeta functions.

## Main theorems #

`LFunction_eq_LSeries`

: if`1 < re s`

then the`LFunction`

coincides with the naive`LSeries`

.`differentiable_LFunction`

: if`χ`

is nontrivial then`LFunction χ s`

is differentiable everywhere.

The unique meromorphic function `ℂ → ℂ`

which agrees with `∑' n : ℕ, χ n / n ^ s`

wherever the
latter is convergent. This is constructed as a linear combination of Hurwitz zeta functions.

Note that this is not the same as `LSeries χ`

: they agree in the convergence range, but
`LSeries χ s`

is defined to be `0`

if `re s ≤ 1`

.

## Equations

- χ.LFunction s = ZMod.LFunction (⇑χ) s

## Instances For

The L-function of the (unique) Dirichlet character mod 1 is the Riemann zeta function.
(Compare `DirichletCharacter.LSeries_modOne_eq`

.)

The L-function of a Dirichlet character is differentiable, except at `s = 1`

if the character is
trivial.

The L-function of a non-trivial Dirichlet character is differentiable everywhere.