LMFDB - L-function 2-1800-3.2-c2-0-16

Dirichlet series

L(s) = 1

− 12·7^-s + 5.65i·11^-s + 8·13^-s + 9.89i·17^-s − 16·19^-s − 39.5i·23^-s + 29.6i·29^-s − 4·31^-s − 30·37^-s + 21.2i·41^-s + 8·43^-s − 16.9i·47^-s + 95·49^-s + 49.4i·53^-s − 79.1i·59^-s + ⋯

L(s) = 1

− 1.71·7^-s + 0.514i·11^-s + 0.615·13^-s + 0.582i·17^-s − 0.842·19^-s − 1.72i·23^-s + 1.02i·29^-s − 0.129·31^-s − 0.810·37^-s + 0.517i·41^-s + 0.186·43^-s − 0.361i·47^-s + 1.93·49^-s + 0.933i·53^-s − 1.34i·59^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1800$ = $2^{3} \cdot 3^{2} \cdot 5^{2}$
Sign:	$0.816 + 0.577i$
Analytic conductor:	$49.0464$
Root analytic conductor:	$7.00331$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1800} (1601, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1800,\ (\ :1),\ 0.816 + 0.577i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$1.147349150$
$L(\frac12)$	$\approx$	$1.147349150$
$L(2)$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	2	$ 1 $
	3	$ 1 $
	5	$ 1 $
good	7	$ 1 + 12T + 49T^{2} $
	11	$ 1 - 5.65iT - 121T^{2} $
	13	$ 1 - 8T + 169T^{2} $
	17	$ 1 - 9.89iT - 289T^{2} $
	19	$ 1 + 16T + 361T^{2} $
	23	$ 1 + 39.5iT - 529T^{2} $
	29	$ 1 - 29.6iT - 841T^{2} $
	31	$ 1 + 4T + 961T^{2} $
	37	$ 1 + 30T + 1.36e3T^{2} $
	41	$ 1 - 21.2iT - 1.68e3T^{2} $
	43	$ 1 - 8T + 1.84e3T^{2} $
	47	$ 1 + 16.9iT - 2.20e3T^{2} $
	53	$ 1 - 49.4iT - 2.80e3T^{2} $
	59	$ 1 + 79.1iT - 3.48e3T^{2} $
	61	$ 1 + 14T + 3.72e3T^{2} $
	67	$ 1 - 88T + 4.48e3T^{2} $
	71	$ 1 - 28.2iT - 5.04e3T^{2} $
	73	$ 1 - 80T + 5.32e3T^{2} $
	79	$ 1 - 100T + 6.24e3T^{2} $
	83	$ 1 + 130. iT - 6.88e3T^{2} $
	89	$ 1 + 148. iT - 7.92e3T^{2} $
	97	$ 1 - 112T + 9.40e3T^{2} $
show more
show less

$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−8.949872704204827714700289255191, −8.451215753540555041945232245442, −7.25152391484621177875145779611, −6.47753819699215696414857661511, −6.12896120240089887751126101168, −4.86405469401109114292111415473, −3.87608706705490292518783526924, −3.12199691012629157347252982505, −2.02599374104832619337668115216, −0.44697356651983851323488964484, 0.71456126264952008921509208088, 2.29034849278222339716352264002, 3.39225219192078984502065546845, 3.86158570093490120323076385879, 5.27675555568343131596046515639, 6.07759120534063402583011707313, 6.65937777876911158136201960911, 7.51529901325894379226546854422, 8.490967396595988469006496639713, 9.309233090623085487202089779698

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(1.147349150\)
\(L(\frac12)\)	\(\approx\)	\(1.147349150\)
\(L(2)\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more