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<h1 align=center> Dirichlet L-functions
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Dirichlet L-functions are Dirichlet series of the form
$$L(s,\chi)=\sum_{n=1}^\infty \frac {\chi(n)}{n^s}$$
where $\chi$ is a primitive Dirichlet character to a modulus $q$.
Equivalently, $\chi$ is a function from the natural numbers to the complex numbers, which is periodic with period $q$ (i.e. $\chi(n+q)=\chi(n)$ for all 
$n\ge 1$), completely multiplicative (i.e. $\chi(mn)=\chi(m)\chi(n)$ for all natural numbers $m$ and $n$), which vanishes at natural numbers which have a factor $> 1$  in common with $q$ and which do not satisfy 
$\chi(m+q_1)=\chi(m)$ for all numbers $m , q_1$ with $q_1<q$ and $(m,q_1)=1$.
The last condition gives the primitivity. Also, we do not consider the function which is identically 0  to be a character. 

If the modulus $q=1$ then $\chi(n)=1$ for all $n$ and $L(s,\chi)=\zeta(s)$. 
If $q>1$, then the series converges for all $s$ with $\sigma >0$ and 
converges absolutely for $\sigma>0$.

$L(s,\chi)$ has an Euler product 
$$L(s,\chi)=\prod _p \left(1-\frac {\chi)p}{p^s}\right)^{-1}.$$

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Back to the 
<a href="http://www.aimath.org/WWN/rh/"> main index</a> 
for The Riemann Hypothesis.
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