Dirichlet L-Functions and Their Zeros

start x axis at (from -100 to 100)

-10

length of x axis (from 1 to 20)

20

modulus (from 1 to 10)

7

character number

4

plot

real and imaginary parts

absolute value

display the zeros in a table

zeros at t =

-6.84549

-4.47574

4.47574

6.84549

Dirichlet

L

-functions are important in number theory. For example,

L

-functions are used to prove Dirichlet's theorem, which states that the arithmetic progression

qn+a

(

n=0,1,2,…

) contains infinitely many primes, provided

q

and

a

are relatively prime. The zeros of

L

-functions can even be used to count how many primes less than

x

there are in arithmetic progressions.

This Demonstration graphs Dirichlet

L

-functions along the line

1/2+it

in the complex plane (the so-called "critical line"), and highlights the zeros that are encountered. Zeros occur where the real part (blue graph) and the imaginary part (red graph) are simultaneously 0.