Dirichlet L-functions
Contents
Definitions
Special case: L-functions with real quadratic character
Dirichlet Series & Euler Product
The Dirichlet L-function, L(s,\chi), has Dirichlet series and Euler Product given by
where \chi is a Dirichlet Character.
Dirichlet Characters
Dirichlet Characters are functions \chi:\mathbb{Z}\rightarrow \mathbb{C} having the following properties:
\chi(n) is a completely multiplicative arithmetic function. That is \chi(mn)=\chi(m)\chi(n) for all m,n\in \mathbb{Z}.
\chi(n) is periodic modulo a positive integer q (called the modulus of \chi). That is \chi(n+rq)=\chi(n) for any n,r\in \mathbb(Z).
\chi(n)=0 if and only if \gcd(n,q)>1
A character \chi is said to be even if \chi(-1)=1 and odd if \chi(-1)=-1
Moreover, \chi must be primitive which means that there does exist another character \chi_1 to a smaller modulus q_1 such that \chi(n)=\chi_1(n) for all n with \gcd(n,q)=1.
Functional Equation
The Functional Equation for L(s,\chi) takes different forms depending on the value of \chi(-1).
Given the completed L-function
the functional equation is given by
where
\tau(\chi) is the Gauss sum, defined as
Zeros
Trivial zeros of L(s,\chi) are at 0 and the negative even integers if \chi is even and at the negative odd integers if \chi is odd.
Special values are at the positive even integers and the negative odd integers for even \chi and positive odd and negative even integers if \chi is odd.
Moments
Moments in t-aspect