§25.15(i) Definitions and Basic Properties

The notation $\mathop{L\/}\nolimits\!\left(s,\chi\right)$ was introduced by Dirichlet (1837) for the meromorphic continuation of the function defined by the series

 25.15.1 $\mathop{L\/}\nolimits\!\left(s,\chi\right)=\sum_{n=1}^{\infty}\frac{\chi(n)}{n% ^{s}},$ $\realpart{s}>1$, Defines: $\mathop{L\/}\nolimits\!\left(s,\chi\right)$: Dirichlet $\mathop{L\/}\nolimits$-function Symbols: $\realpart{}$: real part, $n$: nonnegative integer, $s$: complex variable and $\chi(n)$: Dirichlet character Permalink: http://dlmf.nist.gov/25.15.E1 Encodings: TeX, pMML, png

where $\chi(n)$ is a Dirichlet character $\;\;(\mathop{{\rm mod}}k)$27.8). For the principal character $\chi_{1}\;\;(\mathop{{\rm mod}}k)$, $\mathop{L\/}\nolimits\!\left(s,\chi_{1}\right)$ is analytic everywhere except for a simple pole at $s=1$ with residue $\mathop{\phi\/}\nolimits\!\left(k\right)/k$, where $\mathop{\phi\/}\nolimits\!\left(k\right)$ is Euler’s totient function (§27.2). If $\chi\neq\chi_{1}$, then $\mathop{L\/}\nolimits\!\left(s,\chi\right)$ is an entire function of $s$.

 25.15.2 $\mathop{L\/}\nolimits\!\left(s,\chi\right)=\prod_{p}\left(1-\frac{\chi(p)}{p^{% s}}\right)^{-1},$ $\realpart{s}>1$,

with the product taken over all primes $p$, beginning with $p=2$. This implies that $\mathop{L\/}\nolimits\!\left(s,\chi\right)\neq 0$ if $\realpart{s}>1$.

Equations (25.15.3) and (25.15.4) hold for all $s$ if $\chi\neq\chi_{1}$, and for all $s$ ($\neq 1$) if $\chi=\chi_{1}$:

 25.15.3 $\displaystyle\mathop{L\/}\nolimits\!\left(s,\chi\right)$ $\displaystyle=k^{-s}\sum_{r=1}^{k-1}\chi(r)\mathop{\zeta\/}\nolimits\!\left(s,% \frac{r}{k}\right),$ 25.15.4 $\displaystyle\mathop{L\/}\nolimits\!\left(s,\chi\right)$ $\displaystyle=\mathop{L\/}\nolimits\!\left(s,\chi_{0}\right)\prod_{p\divides k% }\left(1-\frac{\chi_{0}(p)}{p^{s}}\right),$

where $\chi_{0}$ is a primitive character (mod $d$) for some positive divisor $d$ of $k$27.8).

When $\chi$ is a primitive character (mod $k$) the $\mathop{L\/}\nolimits$-functions satisfy the functional equation:

 25.15.5 $\mathop{L\/}\nolimits\!\left(1-s,\chi\right)=\frac{k^{s-1}\mathop{\Gamma\/}% \nolimits\!\left(s\right)}{(2\pi)^{s}}\*{\left(e^{-\pi is/2}+\chi(-1)e^{\pi is% /2}\right)}\*G(\chi)\mathop{L\/}\nolimits\!\left(s,\overline{\chi}\right),$

where $\overline{\chi}$ is the complex conjugate of $\chi$, and

 25.15.6 $G(\chi)=\sum_{r=1}^{k}\chi(r)e^{2\pi ir/k}.$

§25.15(ii) Zeros

Since $\mathop{L\/}\nolimits\!\left(s,\chi\right)\neq 0$ if $\realpart{s}>1$, (25.15.5) shows that for a primitive character $\chi$ the only zeros of $\mathop{L\/}\nolimits\!\left(s,\chi\right)$ for $\realpart{s}<0$ (the so-called trivial zeros) are as follows:

 25.15.7 $\mathop{L\/}\nolimits\!\left(-2n,\chi\right)=0\text{ if }\chi(-1)=1,$ $n=0,1,2,\dots$,
 25.15.8 $\mathop{L\/}\nolimits\!\left(-2n-1,\chi\right)=0\text{ if }\chi(-1)=-1,$ $n=0,1,2,\dots$.

There are also infinitely many zeros in the critical strip $0\leq\realpart{s}\leq 1$, located symmetrically about the critical line $\realpart{s}=\frac{1}{2}$, but not necessarily symmetrically about the real axis.

 25.15.9 $\mathop{L\/}\nolimits\!\left(1,\chi\right)\neq 0\text{ if }\chi\neq\chi_{1},$

where $\chi_{1}$ is the principal character $\;\;(\mathop{{\rm mod}}k)$. This result plays an important role in the proof of Dirichlet’s theorem on primes in arithmetic progressions (§27.11). Related results are:

 25.15.10 $\mathop{L\/}\nolimits\!\left(0,\chi\right)=\begin{cases}\displaystyle-\frac{1}% {k}\sum_{r=1}^{k}r\chi(r),&\chi\neq\chi_{1},\\ 0,&\chi=\chi_{1}.\end{cases}$