# nLab Dirichlet L-function

### Context

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in physics

#### Arithmetic geometry

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# Contents

## Idea

The Dirichlet L-functions $L_\chi$ are a kind of L-function induced by Dirichlet characters $\chi$

$L_\chi(s) \coloneqq \underoverset{n=1}{\infty}{\sum} \chi(n)n^{-s} \,.$

The completion of this by a Gamma function factor (and a power of $\pi$ and of the conductor) is the Mellin transform of the Dirichlet theta function (e.g. Continuations, p. 8).

By Artin reciprocity Dirichlet L-function are equal to suitable Artin L-functions induced by 1-dimensional Galois representations.

In terms of the adelic integral representation of L-functions via Iwasawa-Tate theory, given a Dirichlet character $\chi$ then the corresponding Dirichlet L-functions are simply the adelic integrals (e.g. Garrett 11, section 2.2)

$s \mapsto \int_{\mathbb{I}} \chi(x) \;f(x)\; {\vert x\vert}^s$

for suitable Schwartz functions $f$ on the idele group.

Hence Dirichlet L-functions are Mellin transforms of suitable theta functions (“theta kernels”), see. e.g. (Stopple, p. 3).

Where Dirichlet characters (see there) are essentially automorphic forms for the idele group (hence for $n = 1$) the generalization to automorphic forms for $n \geq 1$ is the concept of automorphic L-function.

The differential geometric analog of a Dirichlet-L function is the eta function (see there for more) of a differential operator.

context/ $\theta$ $\zeta$ (= of $\theta(0,-)$) $L_{\mathbf{z}}$ (= of $\theta(\mathbf{z},-)$) $\eta$
/ $\theta(\mathbf{z},\mathbf{\tau}) = Tr(\exp(-\mathbf{\tau} \cdot (D_\mathbf{z})^2))$ as function of $\mathbf{\tau}$ of $\Sigma$ (hence of ) and / $\mathbf{z}$analytically continued of $\zeta(s) = Tr_{reg}\left(\frac{1}{(D_{0})^2}\right)^s = \int_{0}^\infty \tau^{s-1} \,\theta(0,\tau)\, d\tau$analytically continued of in $\mathbf{z}$: $L_{\mathbf{z}}(s) \coloneqq Tr_{reg}\left(\frac{1}{(D_{\mathbf{z}})^2}\right)^s = \int_{0}^\infty \tau^{s-1} \,\theta(\mathbf{z},\tau)\, d\tau$analytically continued of in $\mathbf{z}$ $\eta_{\mathbf{z}}(s) = Tr_{reg} \left(\frac{sgn(D_{\mathbf{z}})}{ { \vert D_{\mathbf{z}} } \vert }\right)^s$ $pv\, L_{\mathbf{z}}(1) = Tr_{reg}\left(\frac{1}{(D_{\mathbf{z}})^2}\right)$ / fermionic $pv\, \eta_{\mathbf{z}}(1)= Tr_{reg} \left( \frac{D_{\mathbf{z}}}{(D_{\mathbf{z}})^2} \right)$ / $-\frac{1}{2}L_{\mathbf{z}}^\prime(0) = Z_H = \frac{1}{2}\ln\;det_{reg}(D_{\mathbf{z}}^2)$
(),
$\theta(\mathbf{z},\mathbf{\tau})$ of over $J(\Sigma_{\mathbf{\tau}})$ in terms of covering coordinates $\mathbf{z}$ on $\mathbb{C}^g \to J(\Sigma_{\mathbf{\tau}})$
for a (for ) and (in )
for a , (being the $L_{\mathbf{z}}$ for $\mathbf{z} = 0$ the ) $L_{\mathbf{z}}$ of a $\mathbf{z}$, expressible “in coordinates” (by ) as a finite-order (for 1-dimensional representations) and generally (via ) by an (for higher dimensional reps) $\cdot$
for $\mathbb{Q}$ ($\mathbf{z} = 0$)/ ($\mathbf{z} = \chi$ a ) (being the $L_{\mathbf{z}}$ for $\mathbf{z} = 0$) of a $\mathbf{z}$ , expressible “in coordinates” (via ) as a (for 1-dimensional Galois representations) and generally (via ) as an

## References

• E. Kowalski, section 1.3 of Automorphic forms, L-functions and number theory (March 12–16) Three Introductory lectures (pdf)

• section 3 of Continuations and functional equations (pdf)

• Wikipedia, Dirichlet L-function

• Dorian Goldfeld, Joseph Hundley, chapter 2 of Automorphic representations and L-functions for the general linear group, Cambridge Studies in Advanced Mathematics 129, 2011 (pdf)

• Paul Garrett, section 1.6 Iwasawa-Tate on ζ-functions and L-functions

(pdf)

• Jeffrey Stopple, Theta and $L$-function splittings, Acta Arithmetica LXXII.2 (1995) (pdf)

Last revised on July 18, 2015 at 04:23:15. See the history of this page for a list of all contributions to it.