Functional Equation for Dirichlet L-Functions
Jump to navigation
Jump to search
Theorem
Let $\chi$ be a primitive Dirichlet character to the modulus $q \geq 1$.
Let $\map \Lambda {s, \chi}$ be the completed $L$-function for $\chi$.
Let $\map \tau \chi$ denote the Gaussian sum.
Then for all $s \in \C$:
- $\map \Lambda {s, \chi} = i^{-\kappa} \dfrac {\map \tau \chi} {\sqrt q} \map \Lambda {1 - s, \overline \chi}$
where $\kappa = \dfrac 1 2 \paren {1 - \map \chi {-1} }$.
Proof
This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |