LMFDB - L-function 2-12e3-12.11-c3-0-95

Dirichlet series

L(s) = 1

− 14.9i·5^-s − 30.0i·7^-s − 55.9·11^-s − 57.4·13^-s − 29.2i·17^-s − 0.709i·19^-s + 48.0·23^-s − 97.6·25^-s − 172. i·29^-s − 45.2i·31^-s − 448.·35^-s − 248.·37^-s − 51.3i·41^-s + 19.9i·43^-s − 10.8·47^-s + ⋯

L(s) = 1

− 1.33i·5^-s − 1.62i·7^-s − 1.53·11^-s − 1.22·13^-s − 0.417i·17^-s − 0.00856i·19^-s + 0.435·23^-s − 0.781·25^-s − 1.10i·29^-s − 0.262i·31^-s − 2.16·35^-s − 1.10·37^-s − 0.195i·41^-s + 0.0708i·43^-s − 0.0335·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1728$ = $2^{6} \cdot 3^{3}$
Sign:	$-i$
Analytic conductor:	$101.955$
Root analytic conductor:	$10.0972$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1728} (1727, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1728,\ (\ :3/2),\ -i)$

Particular Values

$L(2)$	$\approx$	$0.5393002518$
$L(\frac12)$	$\approx$	$0.5393002518$
$L(\frac{5}{2})$		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 + 14.9iT - 125T^{2} $
	7	$ 1 + 30.0iT - 343T^{2} $
	11	$ 1 + 55.9T + 1.33e3T^{2} $
	13	$ 1 + 57.4T + 2.19e3T^{2} $
	17	$ 1 + 29.2iT - 4.91e3T^{2} $
	19	$ 1 + 0.709iT - 6.85e3T^{2} $
	23	$ 1 - 48.0T + 1.21e4T^{2} $
	29	$ 1 + 172. iT - 2.43e4T^{2} $
	31	$ 1 + 45.2iT - 2.97e4T^{2} $
	37	$ 1 + 248.T + 5.06e4T^{2} $
	41	$ 1 + 51.3iT - 6.89e4T^{2} $
	43	$ 1 - 19.9iT - 7.95e4T^{2} $
	47	$ 1 + 10.8T + 1.03e5T^{2} $
	53	$ 1 + 37.0iT - 1.48e5T^{2} $
	59	$ 1 + 411.T + 2.05e5T^{2} $
	61	$ 1 - 308.T + 2.26e5T^{2} $
	67	$ 1 - 113. iT - 3.00e5T^{2} $
	71	$ 1 + 1.13e3T + 3.57e5T^{2} $
	73	$ 1 - 728.T + 3.89e5T^{2} $
	79	$ 1 + 487. iT - 4.93e5T^{2} $
	83	$ 1 - 1.16e3T + 5.71e5T^{2} $
	89	$ 1 + 1.19e3iT - 7.04e5T^{2} $
	97	$ 1 - 624.T + 9.12e5T^{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

−8.231219609826929663047893795013, −7.58713099388068783529735226379, −7.07627585897985019127370818220, −5.72147100644499046636781091998, −4.79010096966122508218614906467, −4.54865749266276795537953970074, −3.27344223655813610976741756458, −2.04976790867658333803546510200, −0.73422078957385407897000041060, −0.15091497936038621598552706081, 2.04636545459466938906964684763, 2.68852055997650473030819038337, 3.28550018786421685859028187558, 4.95766643419863959191130204360, 5.42440748401289727185393133018, 6.38571785556290872143454462969, 7.17535630109258035623534746884, 7.909377019956496653347939621815, 8.756419868897597851307345748344, 9.578289828119685990541234589139

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.5393002518\)
\(L(\frac12)\)	\(\approx\)	\(0.5393002518\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more