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  "body": "\n## Definition\n\nLet $q \\ge 1$ be an integer and let $\\chi$ be a [[Dirichlet character|7XB5VJKDPG]] modulo $q$.\nFor $s \\in \\C$ with $\\operatorname{Re} s \u003e 1$, the [[Dirichlet series|8MWTEF6462]]\n$$L(s, \\chi) = \\sum_{n=1}^{\\infty} \\frac{\\chi(n)}{n^s}$$\nconverges absolutely and defines a holomorphic function of $s$.\nThe *Dirichlet L-function* of $\\chi$ is the [[analytic continuation|0M3YQJ81NB]] of this function to a [[meromorphic function|0ZZ7ZYDR8W]] on $\\C$, also written $L(s, \\chi)$.\nIf $\\chi$ is not principal, $L(s, \\chi)$ is an [[entire function|0G4F0XB9X2]].\nIf $\\chi = \\chi_0$ is principal, $L(s, \\chi_0)$ has a single pole, a simple pole at $s = 1$.\n\n## Remarks\n\nFor $\\operatorname{Re} s \u003e 1$ there is an Euler product $L(s, \\chi) = \\prod_p (1 - \\chi(p) p^{-s})^{-1}$ over the primes $p$.\nFor the principal character, $L(s, \\chi_0) = \\zeta(s) \\prod_{p \\mid q} (1 - p^{-s})$, where $\\zeta$ is the [[Riemann zeta function|1XWBY4N0VT]].\nExample: for the nonprincipal character $\\chi$ modulo $4$, the series at $s = 1$ converges and $L(1, \\chi) = 1 - \\frac{1}{3} + \\frac{1}{5} - \\cdots = \\frac{\\pi}{4}$, by the [[Leibniz formula for pi|037FD5NBRT]].\nNon-example: $\\sum_{n} (-1)^{n+1} n^{-s}$ is a Dirichlet series with $2$-periodic coefficients, but it is not a Dirichlet L-function, since $f(n) = (-1)^{n+1}$ is not a Dirichlet character: $f(4) = -1 \\ne 1 = f(2) f(2)$.\nFor a primitive character the completed L-function satisfies a functional equation relating $s$ and $1 - s$, as for the zeta function.\n",
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