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  "body": "\n## Summary\n\nLet $K$ be an [[algebraic number field|MWBDFS2KJ5]].\nIts Dedekind zeta function $\\zeta_K(s)$ is the sum of $N(\\mathfrak{a})^{-s}$ over the nonzero ideals $\\mathfrak{a}$ of the ring of integers.\nFor $K = \\Q$ it is the [[Riemann zeta function|1XWBY4N0VT]].\nIt has an Euler product over the prime ideals and a functional equation.\nIt extends to a meromorphic function on the whole complex plane with a single simple pole at $s = 1$.\nIts residue at $s = 1$ is given by the [[class number formula|55XX5A7XZV]].\nThis formula ties together the class number, the [[regulator|RWK0HNF1NA]] and the [[discriminant|E0VZTNPJXA]].\n\n## Explanations\n\n### L1 human\n\nThe Riemann zeta function adds up $1/n^s$ over the positive integers.\nThe Dedekind zeta function does the same for a number field, with ideals in place of integers and their norms in place of $n$.\nFor the Gaussian integers, the ideals of norm $5$ are $(2 + i)$ and $(2 - i)$, so the term $5^{-s}$ appears twice.\nThe analytic behaviour of this function encodes how the prime ideals of the field are distributed.\n",
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