LMFDB - L-function 2-1840-92.91-c1-0-24

Dirichlet series

L(s) = 1

− 0.431i·3^-s − i·5^-s − 3.33·7^-s + 2.81·9^-s − 1.49·11^-s + 5.60·13^-s − 0.431·15^-s + 0.587i·17^-s + 0.133·19^-s + 1.43i·21^-s + (3.96 + 2.70i)23^-s − 25^-s − 2.51i·27^-s − 1.54·29^-s − 1.79i·31^-s + ⋯

L(s) = 1

− 0.249i·3^-s − 0.447i·5^-s − 1.25·7^-s + 0.937·9^-s − 0.450·11^-s + 1.55·13^-s − 0.111·15^-s + 0.142i·17^-s + 0.0306·19^-s + 0.313i·21^-s + (0.825 + 0.563i)23^-s − 0.200·25^-s − 0.483i·27^-s − 0.286·29^-s − 0.321i·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.433 + 0.901i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.433 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1840$ = $2^{4} \cdot 5 \cdot 23$
Sign:	$0.433 + 0.901i$
Analytic conductor:	$14.6924$
Root analytic conductor:	$3.83307$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1840} (1471, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1840,\ (\ :1/2),\ 0.433 + 0.901i)$

Particular Values

$L(1)$	$\approx$	$1.574031760$
$L(\frac12)$	$\approx$	$1.574031760$
$L(\frac{3}{2})$		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 $
	5	$ 1 + iT $
	23	$ 1 + (-3.96 - 2.70i)T $
good	3	$ 1 + 0.431iT - 3T^{2} $
	7	$ 1 + 3.33T + 7T^{2} $
	11	$ 1 + 1.49T + 11T^{2} $
	13	$ 1 - 5.60T + 13T^{2} $
	17	$ 1 - 0.587iT - 17T^{2} $
	19	$ 1 - 0.133T + 19T^{2} $
	29	$ 1 + 1.54T + 29T^{2} $
	31	$ 1 + 1.79iT - 31T^{2} $
	37	$ 1 + 5.62iT - 37T^{2} $
	41	$ 1 + 1.87T + 41T^{2} $
	43	$ 1 - 5.67T + 43T^{2} $
	47	$ 1 + 6.73iT - 47T^{2} $
	53	$ 1 - 0.671iT - 53T^{2} $
	59	$ 1 + 3.46iT - 59T^{2} $
	61	$ 1 + 13.7iT - 61T^{2} $
	67	$ 1 - 3.10T + 67T^{2} $
	71	$ 1 + 12.2iT - 71T^{2} $
	73	$ 1 - 15.3T + 73T^{2} $
	79	$ 1 + 8.78T + 79T^{2} $
	83	$ 1 - 11.3T + 83T^{2} $
	89	$ 1 + 1.49iT - 89T^{2} $
	97	$ 1 - 1.91iT - 97T^{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

−9.236604425223408798895746071857, −8.340204888643283058384925640642, −7.49943991147383719096290644298, −6.67125942982020017247917315802, −6.03887092371551360505576691688, −5.10928347143126275347566382130, −3.93231681120156240360406330043, −3.33730500243176400955722457631, −1.92770054758787869271947524396, −0.69923762420174064532279115556, 1.12926486325409796627501428871, 2.69222374942196819653989377588, 3.50236886525010431081341039022, 4.27051976753118522978871996427, 5.43654288674154602380223375961, 6.38942890036410876489046000067, 6.83092378365997501933882407274, 7.75779515135123366756947603115, 8.762692528146680185998374549314, 9.426096178169565780550326143445

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.574031760\)
\(L(\frac12)\)	\(\approx\)	\(1.574031760\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more