LMFDB - L-function 2-3024-28.27-c1-0-39

Dirichlet series

L(s) = 1

− 2.18i·5^-s + (1 − 2.44i)7^-s + 5.26i·11^-s + 1.01i·13^-s − 3.08i·17^-s + 3.24·19^-s + 0.903i·23^-s + 0.242·25^-s + 5.34·29^-s + 5.24·31^-s + (−5.34 − 2.18i)35^-s + 37^-s + 8.35i·41^-s − 4.47i·43^-s + 12.8·47^-s + ⋯

L(s) = 1

− 0.975i·5^-s + (0.377 − 0.925i)7^-s + 1.58i·11^-s + 0.281i·13^-s − 0.748i·17^-s + 0.743·19^-s + 0.188i·23^-s + 0.0485·25^-s + 0.992·29^-s + 0.941·31^-s + (−0.903 − 0.368i)35^-s + 0.164·37^-s + 1.30i·41^-s − 0.682i·43^-s + 1.88·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.612 + 0.790i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.612 + 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3024$ = $2^{4} \cdot 3^{3} \cdot 7$
Sign:	$0.612 + 0.790i$
Analytic conductor:	$24.1467$
Root analytic conductor:	$4.91393$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3024} (1567, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3024,\ (\ :1/2),\ 0.612 + 0.790i)$

Particular Values

$L(1)$	$\approx$	$2.012418122$
$L(\frac12)$	$\approx$	$2.012418122$
$L(\frac{3}{2})$		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 $
	3	$ 1 $
	7	$ 1 + (-1 + 2.44i)T $
good	5	$ 1 + 2.18iT - 5T^{2} $
	11	$ 1 - 5.26iT - 11T^{2} $
	13	$ 1 - 1.01iT - 13T^{2} $
	17	$ 1 + 3.08iT - 17T^{2} $
	19	$ 1 - 3.24T + 19T^{2} $
	23	$ 1 - 0.903iT - 23T^{2} $
	29	$ 1 - 5.34T + 29T^{2} $
	31	$ 1 - 5.24T + 31T^{2} $
	37	$ 1 - T + 37T^{2} $
	41	$ 1 - 8.35iT - 41T^{2} $
	43	$ 1 + 4.47iT - 43T^{2} $
	47	$ 1 - 12.8T + 47T^{2} $
	53	$ 1 + 7.55T + 53T^{2} $
	59	$ 1 - 5.34T + 59T^{2} $
	61	$ 1 + 3.46iT - 61T^{2} $
	67	$ 1 - 3.88iT - 67T^{2} $
	71	$ 1 - 2.18iT - 71T^{2} $
	73	$ 1 + 14.2iT - 73T^{2} $
	79	$ 1 + 9.37iT - 79T^{2} $
	83	$ 1 + 12.8T + 83T^{2} $
	89	$ 1 + 3.98iT - 89T^{2} $
	97	$ 1 + 6.33iT - 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

−8.624356355037225521473805711270, −7.75772655225807916400513960961, −7.24961733922511828458188219036, −6.49782564338728490722398886399, −5.26070610997266333498647939629, −4.64374306709071557329941279827, −4.23500335880941127213608801185, −2.91175835356353368399335585846, −1.69354537167868441863769487856, −0.805454613890060925437154469751, 1.03064429535488964589658588265, 2.52390058660053924227946943497, 3.02104588870076831572568379640, 3.96982527596415849892598821957, 5.17793812250280011160131129726, 5.88506579742064536606287400488, 6.41062398081760844473146472813, 7.33820438372341476327529592924, 8.299308976561922338931248660943, 8.581978989689940955989537564330

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.012418122\)
\(L(\frac12)\)	\(\approx\)	\(2.012418122\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more