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  "body": "\n## Summary\n\nA Hecke character of a [[number field|MWBDFS2KJ5]] $K$ is a continuous homomorphism from the [[idele class group|9GHD9MXNYV]] $C_K$ to the nonzero complex numbers.\nHecke characters generalize [[Dirichlet characters|7XB5VJKDPG]], which are the Hecke characters of finite order of $\\Q$.\nEach Hecke character has an L-function with an Euler product, a meromorphic continuation and a functional equation, proved by Hecke and again by [[Tate's thesis|QYW79XKJTP]].\n\n## Explanations\n\n### L1 human\n\nA Dirichlet character is a pattern of complex numbers attached to the residues modulo $m$, compatible with multiplication.\nA Hecke character does the same for a number field, but it may also depend on the real and complex embeddings in a continuous way.\nIn the language of ideles it is just a continuous character of the idele group that ignores the elements of $K$.\nA simple example is the idele norm $x \\mapsto \\prod_v \\abs{x_v}_v$, which is $1$ on every element of $K$ by the product formula.\n",
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