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  "body": "\n## Summary\n\nThe Hurwitz zeta function $\\zeta(s, a)$ is the sum of $(n + a)^{-s}$ over all integers $n \\ge 0$.\nIt is a shifted version of the [[Riemann zeta function|1XWBY4N0VT]], which is the case $a = 1$.\nEvery [[Dirichlet L-function|9D2Z543EJQ]] modulo $q$ is $q^{-s}$ times a finite linear combination of the functions $\\zeta(s, r/q)$, so their analytic properties follow from those of $\\zeta(s, a)$.\n\n## Explanations\n\n### L1 human\n\nThe Riemann zeta function adds up $1/n^s$ for $n = 1, 2, 3, \\ldots$.\nThe Hurwitz zeta function does the same, but starts the counting at a point $a$ instead of at $1$: it adds $1/a^s$, $1/(a+1)^s$, $1/(a+2)^s$, and so on.\nFor $a = 1$ you get back the Riemann zeta function.\nFor $a = 1/2$ you add the $s$-th powers of $2, 2/3, 2/5, \\ldots$, and this sum equals $(2^s - 1)\\zeta(s)$.\nSums over an arithmetic progression, such as $1/1^s + 1/4^s + 1/7^s + \\cdots$, are Hurwitz zeta values up to a factor.\n",
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