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  "body": "\n## Summary\n\nA Dirichlet L-function is the [[Dirichlet series|8MWTEF6462]] $L(s, \\chi) = \\sum_{n \\ge 1} \\chi(n) n^{-s}$ of a [[Dirichlet character|7XB5VJKDPG]] $\\chi$, together with its analytic continuation.\nIt has an Euler product over the primes and generalizes the [[Riemann zeta function|1XWBY4N0VT]], which is the case of the trivial character modulo $1$.\nThe fact that $L(1, \\chi) \\ne 0$ for every nonprincipal $\\chi$ is the heart of [[Dirichlet's theorem on arithmetic progressions|0GNVN3CA2P]].\n\n## Explanations\n\n### L1 human\n\nThe zeta function adds $1/n^s$ over all positive integers.\nA Dirichlet L-function adds the same terms with the weights $\\chi(n)$, which depend on $n$ modulo $q$.\nThese weights let one separate the primes according to their remainder modulo $q$.\nFor the character modulo $4$ with values $1, 0, -1, 0$, the value at $s = 1$ is $1 - 1/3 + 1/5 - 1/7 + \\cdots = \\pi/4$.\n",
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