# §25.15 Dirichlet $L$-functions

## §25.15(i) Definitions and Basic Properties

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

 25.15.1 $L\left(s,\chi\right)=\sum_{n=1}^{\infty}\frac{\chi(n)}{n^{s}},$ $\Re s>1$, ⓘ Defines: $L\left(\NVar{s},\NVar{\chi}\right)$: Dirichlet $L$-function Symbols: $\Re$: real part, $n$: nonnegative integer, $s$: complex variable and $\chi(n)$: Dirichlet character Keywords: definition, infinite series, series representation Source: Apostol (1976, p. 249) Referenced by: §25.11(x), §25.11(x), 3rd Erratum (V1.0.23) Permalink: http://dlmf.nist.gov/25.15.E1 Encodings: TeX, pMML, png See also: Annotations for §25.15(i), §25.15 and Ch.25

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

 25.15.2 $L\left(s,\chi\right)=\prod_{p}\left(1-\frac{\chi(p)}{p^{s}}\right)^{-1},$ $\Re s>1$, ⓘ Symbols: $L\left(\NVar{s},\NVar{\chi}\right)$: Dirichlet $L$-function, $\Re$: real part, $p$: prime number, $s$: complex variable and $\chi(n)$: Dirichlet character Keywords: infinite product, product representation Source: Apostol (1976, p. 231) Permalink: http://dlmf.nist.gov/25.15.E2 Encodings: TeX, pMML, png See also: Annotations for §25.15(i), §25.15 and Ch.25

with the product taken over all primes $p$, beginning with $p=2$. This implies that $L\left(s,\chi\right)\neq 0$ if $\Re s>1$.

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 $\displaystyle L\left(s,\chi\right)$ $\displaystyle=k^{-s}\sum_{r=1}^{k-1}\chi(r)\zeta\left(s,\frac{r}{k}\right),$ ⓘ Symbols: $L\left(\NVar{s},\NVar{\chi}\right)$: Dirichlet $L$-function, $\zeta\left(\NVar{s},\NVar{a}\right)$: Hurwitz zeta function, $k$: nonnegative integer, $s$: complex variable and $\chi(n)$: Dirichlet character Keywords: finite sum Source: Apostol (1976, p. 249) Referenced by: §25.11(x), §25.11(x), §25.15(i), 3rd Erratum (V1.0.23) Permalink: http://dlmf.nist.gov/25.15.E3 Encodings: TeX, pMML, png See also: Annotations for §25.15(i), §25.15 and Ch.25 25.15.4 $\displaystyle L\left(s,\chi\right)$ $\displaystyle=L\left(s,\chi_{0}\right)\prod_{p\mathbin{|}k}\left(1-\frac{\chi_% {0}(p)}{p^{s}}\right),$ ⓘ Symbols: $L\left(\NVar{s},\NVar{\chi}\right)$: Dirichlet $L$-function, $k$: nonnegative integer, $p$: prime number, $s$: complex variable and $\chi(n)$: Dirichlet character Keywords: product representation Source: Apostol (1976, p. 262) Referenced by: §25.15(i) Permalink: http://dlmf.nist.gov/25.15.E4 Encodings: TeX, pMML, png See also: Annotations for §25.15(i), §25.15 and Ch.25

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 $L$-functions satisfy the functional equation:

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

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

 25.15.6 $G(\chi)\equiv\sum_{r=1}^{k}\chi(r)e^{2\pi ir/k}.$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\equiv$: equals by definition, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer and $\chi(n)$: Dirichlet character Keywords: definition Source: Apostol (1976, p. 262) Permalink: http://dlmf.nist.gov/25.15.E6 Encodings: TeX, pMML, png See also: Annotations for §25.15(i), §25.15 and Ch.25

## §25.15(ii) Zeros

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

 25.15.7 $L\left(-2n,\chi\right)=0\text{ if }\chi(-1)=1,$ $n=0,1,2,\dots$, ⓘ Symbols: $L\left(\NVar{s},\NVar{\chi}\right)$: Dirichlet $L$-function, $n$: nonnegative integer and $\chi(n)$: Dirichlet character Keywords: zeros Source: Apostol (1985b, p. 29) Permalink: http://dlmf.nist.gov/25.15.E7 Encodings: TeX, pMML, png See also: Annotations for §25.15(ii), §25.15 and Ch.25
 25.15.8 $L\left(-2n-1,\chi\right)=0\text{ if }\chi(-1)=-1,$ $n=0,1,2,\dots$. ⓘ Symbols: $L\left(\NVar{s},\NVar{\chi}\right)$: Dirichlet $L$-function, $n$: nonnegative integer and $\chi(n)$: Dirichlet character Keywords: zeros Source: Apostol (1985b, p. 29) Permalink: http://dlmf.nist.gov/25.15.E8 Encodings: TeX, pMML, png See also: Annotations for §25.15(ii), §25.15 and Ch.25

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

 25.15.9 $L\left(1,\chi\right)\neq 0\text{ if }\chi\neq\chi_{1},$ ⓘ Symbols: $L\left(\NVar{s},\NVar{\chi}\right)$: Dirichlet $L$-function and $\chi(n)$: Dirichlet character Keywords: zeros Source: Apostol (1976, p. 149) Permalink: http://dlmf.nist.gov/25.15.E9 Encodings: TeX, pMML, png See also: Annotations for §25.15(ii), §25.15 and Ch.25

where $\chi_{1}$ is the principal character $\pmod{k}$. 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 $L\left(0,\chi\right)=\begin{cases}\displaystyle-\frac{1}{k}\sum_{r=1}^{k}r\chi% (r),&\chi\neq\chi_{1},\\ 0,&\chi=\chi_{1}.\end{cases}$ ⓘ Symbols: $L\left(\NVar{s},\NVar{\chi}\right)$: Dirichlet $L$-function, $k$: nonnegative integer and $\chi(n)$: Dirichlet character Keywords: zeros Source: Apostol (1976, Theorem 12.20, p. 268) Permalink: http://dlmf.nist.gov/25.15.E10 Encodings: TeX, pMML, png See also: Annotations for §25.15(ii), §25.15 and Ch.25