LMFDB - L-function 2-2880-3.2-c2-0-48

Dirichlet series

L(s) = 1

− 2.23i·5^-s + 13.4·7^-s + 17.6i·11^-s + 7.48·13^-s − 16.9i·17^-s − 10.9·19^-s − 21.9i·23^-s − 5.00·25^-s − 47.3i·29^-s + 16.9·31^-s − 30.1i·35^-s + 5.53·37^-s − 66.3i·41^-s + 38.9·43^-s + 32.5i·47^-s + ⋯

L(s) = 1

− 0.447i·5^-s + 1.92·7^-s + 1.60i·11^-s + 0.575·13^-s − 0.998i·17^-s − 0.577·19^-s − 0.952i·23^-s − 0.200·25^-s − 1.63i·29^-s + 0.547·31^-s − 0.861i·35^-s + 0.149·37^-s − 1.61i·41^-s + 0.906·43^-s + 0.692i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2880$ = $2^{6} \cdot 3^{2} \cdot 5$
Sign:	$0.816 + 0.577i$
Analytic conductor:	$78.4743$
Root analytic conductor:	$8.85857$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2880} (1601, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2880,\ (\ :1),\ 0.816 + 0.577i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$2.867889650$
$L(\frac12)$	$\approx$	$2.867889650$
$L(2)$		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 + 2.23iT $
good	7	$ 1 - 13.4T + 49T^{2} $
	11	$ 1 - 17.6iT - 121T^{2} $
	13	$ 1 - 7.48T + 169T^{2} $
	17	$ 1 + 16.9iT - 289T^{2} $
	19	$ 1 + 10.9T + 361T^{2} $
	23	$ 1 + 21.9iT - 529T^{2} $
	29	$ 1 + 47.3iT - 841T^{2} $
	31	$ 1 - 16.9T + 961T^{2} $
	37	$ 1 - 5.53T + 1.36e3T^{2} $
	41	$ 1 + 66.3iT - 1.68e3T^{2} $
	43	$ 1 - 38.9T + 1.84e3T^{2} $
	47	$ 1 - 32.5iT - 2.20e3T^{2} $
	53	$ 1 - 11.2iT - 2.80e3T^{2} $
	59	$ 1 - 31.8iT - 3.48e3T^{2} $
	61	$ 1 + 46.9T + 3.72e3T^{2} $
	67	$ 1 + 76T + 4.48e3T^{2} $
	71	$ 1 + 77.7iT - 5.04e3T^{2} $
	73	$ 1 - 94.9T + 5.32e3T^{2} $
	79	$ 1 - 6.92T + 6.24e3T^{2} $
	83	$ 1 - 62.1iT - 6.88e3T^{2} $
	89	$ 1 - 62.2iT - 7.92e3T^{2} $
	97	$ 1 + 124.T + 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

−8.483482445071059778671984421686, −7.74833284459452158654969710913, −7.28258508508349272710591993283, −6.17990598062681430257305928857, −5.23638906771885388915955998380, −4.46658913911784043237332720750, −4.26600508952183869611069059780, −2.44976854521958633413799941586, −1.84623490868083196124558802835, −0.75059489951598764493378784853, 1.06845278028112520599159101511, 1.80824935877262457188246710235, 3.06813700998091811170376018729, 3.89567027962258428786873764231, 4.81234169772479876157869346882, 5.65662858453707675383879941994, 6.24224535159577579323815248261, 7.28818692337533618632995184005, 8.225823261483070022937296397304, 8.346115627721498003924627555254

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(2.867889650\)
\(L(\frac12)\)	\(\approx\)	\(2.867889650\)
\(L(2)\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more