Hecke L-functions

\(\DeclareMathOperator{\diff}{\, d}\) \(\DeclareMathOperator{\vol}{vol}\) \(\DeclareMathOperator{\Res}{Res}\)

Let \(K\) be a number field. Let \(\chi : \mathbb{A}_{K}^{\times}/K^{\times}\) be a Hecke character. Suppose \(\varphi: \mathbb{A}_{K} \rightarrow K\) is Schwartz-Bruhat function. Then, one has the following Tate integral to express a Hecke \(L\)-function.

\[\begin{equation} L(\chi, \varphi ,s) = \int_{\mathbb{A}_{K}^{\times}}^{} \varphi(x) \chi(x) |x|_{\mathbb{A}}^{s}\diff ^{\times}x. \end{equation}\]

Since \(\chi\) is invariant under multiplication by \(K^{\times}\), one can now rewrite this integral as \[\begin{equation} L(\chi,\varphi,s) = \int_{\mathbb{A}_{K}^{\times}/K^{\times}}^{} \Big(\sum_{y \in K^{\times}}^{} \varphi(xy)\Big) \chi(x) |x|^{s}_{\mathbb{A}_{K}^{\times}} \diff ^{\times} x . \end{equation}\]

This allows one to prove the functional equation of the Hecke \(L\)-function. One has to use the Adelic Poisson summation formula to get \[\begin{equation} \sum_{y \in K} \varphi(y) = \sum_{u \in K}^{} \widehat{\varphi}(u) \end{equation}\] This produces the functional equation \[\begin{equation} L(\chi,\varphi , s) = L(\chi^{-1}, \widehat{\varphi} , 1-s). \end{equation}\]

See the notes of Paul Garrett for some more information. In particular, he states that \[\begin{equation} \Res_{s=1} L(\chi, \varphi , s ) = \begin{cases} \vol(\mathbb{A}_{K}^{\times}/K^{\times}) \cdot \widehat{\varphi}(0) & \text{ if } \chi \text{ is trivial }\\ 0 & \text{ otherwise} \end{cases}. \end{equation}\]