---
date: 2022-01-26 15:10
modification date: Wednesday 26th January 2022 15:10:43
title: L function
aliases: [L functions, Dirichlet character, Dirichlet series]

---


- Tags 
	- #arithmetic-geometry/Langlands 
- Refs:
	- Course notes on arithmetic of L functions: <https://www.math.columbia.edu/~phlee/CourseNotes/L-functions.pdf#page=1> #resources/course-notes 
	- Course notes on special values: <https://www.math.columbia.edu/~phlee/CourseNotes/Nonvanishing.pdf#page=2> #resources/course-notes 
	- Course notes on automorphic forms: <https://www.math.columbia.edu/~phlee/CourseNotes/Automorphic.pdf#page=1> #resources/course-notes 
	- [Graduate seminar notes: What is an L function?](https://www.math.leidenuniv.nl/~pbruin/L-functions.pdf#page=1) #resources/summaries 
	- P. Deligne, Valeurs de fonctions L et p ́eriodes d’int ́egrales. 
	  In: A. Borel and W. Casselman (editors), Automorphic Forms, Representations, and L-Functions
- Links:
	- [Dirichlet's theorem on primes in arithmetic progressions](Unsorted/Dirichlet's%20theorem%20on%20primes%20in%20arithmetic%20progressions.md)
	- Types of L functions:
		- [Dedekind zeta function](Unsorted/Dedekind%20zeta%20function.md)
		- [Riemann zeta function](Unsorted/Riemann%20zeta%20function.md)
		- [p-adic zeta function](Unsorted/p-adic%20zeta%20function.md)
		- [motivic zeta function](Unsorted/motivic%20zeta%20function.md)
		- [Artin L function](Artin%20L%20function)
		- [automorphic L function](automorphic%20L%20function)
		- [L function of a modular form](Unsorted/modular%20form.md)
		- [Artin L function](Artin%20L%20function) of a [Galois representation](Unsorted/Galois%20representations.md)
		- [L function of an elliptic curve](MOCs/elliptic%20curve.md)
		- [Hasse-Weil L function](Unsorted/Hasse-Weil%20L%20function.md)

# L function
![](attachments/2023-01-12-7.png)

![](attachments/2023-01-1212321321.png)

![](attachments/Pasted%20image%2020220430211056.png)
![](attachments/Pasted%20image%2020220430211200.png)


![](attachments/Pasted%20image%2020220126151039.png)

## Automorphy

See [automorphic form](Unsorted/automorphic%20form.md) and [automorphic representation](Unsorted/automorphic%20representation.md):
![](attachments/Pasted%20image%2020220430211242.png)

# Characters
![](attachments/Pasted%20image%2020220126151911.png)

![](attachments/Pasted%20image%2020220126153654.png)

## Orthogonality of Characters

![](attachments/Pasted%20image%2020220126154042.png)
![](attachments/Pasted%20image%2020220126154146.png)
# Conductors
![](attachments/Pasted%20image%2020220126152605.png)

![](attachments/Pasted%20image%2020220126152652.png)