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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.
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