LMFDB - L-function 2-864-24.5-c2-0-6

Dirichlet series

L(s) = 1

− 2.62·5^-s − 0.432·7^-s + 5.01·11^-s + 1.56i·13^-s + 19.8i·17^-s − 24.1i·19^-s + 3.07i·23^-s − 18.1·25^-s − 34.4·29^-s + 43.4·31^-s + 1.13·35^-s + 52.9i·37^-s + 56.7i·41^-s − 27.6i·43^-s + 83.7i·47^-s + ⋯

L(s) = 1

− 0.524·5^-s − 0.0617·7^-s + 0.455·11^-s + 0.120i·13^-s + 1.16i·17^-s − 1.27i·19^-s + 0.133i·23^-s − 0.725·25^-s − 1.18·29^-s + 1.40·31^-s + 0.0323·35^-s + 1.43i·37^-s + 1.38i·41^-s − 0.643i·43^-s + 1.78i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.299 - 0.954i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.299 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$864$ = $2^{5} \cdot 3^{3}$
Sign:	$-0.299 - 0.954i$
Analytic conductor:	$23.5422$
Root analytic conductor:	$4.85204$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{864} (593, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 864,\ (\ :1),\ -0.299 - 0.954i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.9910056621$
$L(\frac12)$	$\approx$	$0.9910056621$
$L(2)$		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 $
good	5	$ 1 + 2.62T + 25T^{2} $
	7	$ 1 + 0.432T + 49T^{2} $
	11	$ 1 - 5.01T + 121T^{2} $
	13	$ 1 - 1.56iT - 169T^{2} $
	17	$ 1 - 19.8iT - 289T^{2} $
	19	$ 1 + 24.1iT - 361T^{2} $
	23	$ 1 - 3.07iT - 529T^{2} $
	29	$ 1 + 34.4T + 841T^{2} $
	31	$ 1 - 43.4T + 961T^{2} $
	37	$ 1 - 52.9iT - 1.36e3T^{2} $
	41	$ 1 - 56.7iT - 1.68e3T^{2} $
	43	$ 1 + 27.6iT - 1.84e3T^{2} $
	47	$ 1 - 83.7iT - 2.20e3T^{2} $
	53	$ 1 - 41.4T + 2.80e3T^{2} $
	59	$ 1 + 74.2T + 3.48e3T^{2} $
	61	$ 1 - 28.7iT - 3.72e3T^{2} $
	67	$ 1 - 33.8iT - 4.48e3T^{2} $
	71	$ 1 - 104. iT - 5.04e3T^{2} $
	73	$ 1 - 53.2T + 5.32e3T^{2} $
	79	$ 1 + 51.8T + 6.24e3T^{2} $
	83	$ 1 + 76.3T + 6.88e3T^{2} $
	89	$ 1 - 131. iT - 7.92e3T^{2} $
	97	$ 1 - 68.9T + 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

−10.15876455465654943556235398587, −9.391816765454986133642926912315, −8.462839657042549274289708299780, −7.76373053651081296781971612750, −6.74477611425716132082639062791, −5.99666190029063484807295614751, −4.73107296987061237350594960309, −3.93512738098019111752013660053, −2.80576984171132211887016945051, −1.32776100244021745356465117732, 0.34300000207301719277076094333, 1.94757135261762573288396116563, 3.35311460403231084647447386841, 4.16921538207397100092054440685, 5.30912067049114530773637233370, 6.23449446189124684954903824485, 7.29183547203701587437869299227, 7.914112176272117502670558597585, 8.910715342753473478062448182330, 9.695756503009130333931025446809

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.9910056621\)
\(L(\frac12)\)	\(\approx\)	\(0.9910056621\)
\(L(2)\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more