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I'm working on L-series (reading Rosen's book Number Theory in Function fields) and i read that Dirichlet $L$-series are supposed to be a special case of Hecke $L$-series, and i can't understand why ?

hyuno
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    Dirichlet characters are the (finite order) Hecke characters of $\Bbb{Q}$. https://math.stackexchange.com/a/3230897/276986 – reuns Feb 26 '21 at 10:09
  • I'm sorry i don't understand this post – hyuno Feb 26 '21 at 10:16
  • A Dirichlet character is a periodic completely multiplicative function on the integers. I am showing how to convert it to a continuous homomorphism $\Bbb{A_Q^\times/Q^\times\to C^\times}$, a Hecke character of $\Bbb{Q}$. – reuns Feb 26 '21 at 10:19
  • Did you make any progress? What is a Hecke character to you? – reuns Feb 27 '21 at 06:51
  • I'm working on function fields, i don't have the general definition of a Hecke character : Let $F$ be an effective divisor with support $S \subset S_K$. A character of finite order on the ray class group modulo $F$ is called a Hecke character modulo $F$ – hyuno Feb 27 '21 at 09:25
  • Then it is immediate that this definition works for number fields and that for $\Bbb{Q}$ it gives the Dirichlet characters, right? – reuns Feb 27 '21 at 09:52
  • See http://people.reed.edu/~jerry/361/lectures/heckechar.pdf – Watson May 26 '21 at 19:09

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