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Mathematica > Mathematics and Algorithms > Number Theory > Analytic Number Theory >

Built-in Mathematica Symbol

DirichletL

DirichletL[k, j, s]
gives the Dirichlet L-function

for the Dirichlet character

with modulus k and index j.

The possible Dirichlet characters modulo k are specified by an index j, and given by DirichletCharacter[k, j, n].

DirichletL[k, j, s] can be evaluated to arbitrary numerical precision for integer k and j, and arbitrary complex s.

A plot along the line 1+

A plot along the line 1+

Evaluate for complex arguments:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Simple exact values are generated automatically:

DirichletL threads element-wise over lists and matrices:

Plot the real part of a DirichletL function on the critical line:

Plot the real part across the critical strip:

Find a zero of a DirichletL function:

Plot real and imaginary parts in the vicinity of nearby zeros:

The real part of a Dirichlet L-function: