LMFDB - L-function 2-12e3-3.2-c2-0-48

Dirichlet series

L(s) = 1

− 6.61i·5^-s + 9.43·7^-s + 17.6i·11^-s − 20.6·13^-s − 16.0i·17^-s + 12.2·19^-s − 26.6i·23^-s − 18.6·25^-s − 0.921i·29^-s + 53.7·31^-s − 62.3i·35^-s + 64.0·37^-s − 19.7i·41^-s − 69.7·43^-s − 0.606i·47^-s + ⋯

L(s) = 1

− 1.32i·5^-s + 1.34·7^-s + 1.60i·11^-s − 1.59·13^-s − 0.944i·17^-s + 0.647·19^-s − 1.16i·23^-s − 0.747·25^-s − 0.0317i·29^-s + 1.73·31^-s − 1.78i·35^-s + 1.73·37^-s − 0.482i·41^-s − 1.62·43^-s − 0.0128i·47^-s + ⋯

Functional equation

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

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

Invariants

Degree:	$2$
Conductor:	$1728$ = $2^{6} \cdot 3^{3}$
Sign:	$i$
Analytic conductor:	$47.0845$
Root analytic conductor:	$6.86182$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1728} (1025, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1728,\ (\ :1),\ i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$2.030022195$
$L(\frac12)$	$\approx$	$2.030022195$
$L(2)$		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 + 6.61iT - 25T^{2} $
	7	$ 1 - 9.43T + 49T^{2} $
	11	$ 1 - 17.6iT - 121T^{2} $
	13	$ 1 + 20.6T + 169T^{2} $
	17	$ 1 + 16.0iT - 289T^{2} $
	19	$ 1 - 12.2T + 361T^{2} $
	23	$ 1 + 26.6iT - 529T^{2} $
	29	$ 1 + 0.921iT - 841T^{2} $
	31	$ 1 - 53.7T + 961T^{2} $
	37	$ 1 - 64.0T + 1.36e3T^{2} $
	41	$ 1 + 19.7iT - 1.68e3T^{2} $
	43	$ 1 + 69.7T + 1.84e3T^{2} $
	47	$ 1 + 0.606iT - 2.20e3T^{2} $
	53	$ 1 + 45.3iT - 2.80e3T^{2} $
	59	$ 1 + 71.3iT - 3.48e3T^{2} $
	61	$ 1 - 4T + 3.72e3T^{2} $
	67	$ 1 + 7.43T + 4.48e3T^{2} $
	71	$ 1 + 27.3iT - 5.04e3T^{2} $
	73	$ 1 - 6.30T + 5.32e3T^{2} $
	79	$ 1 + 42.6T + 6.24e3T^{2} $
	83	$ 1 + 72.3iT - 6.88e3T^{2} $
	89	$ 1 + 126. iT - 7.92e3T^{2} $
	97	$ 1 - 27T + 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

−8.936847652665717429791592790522, −8.050029314875880706613784574368, −7.58216528237321194363788815233, −6.68746822408359464462633428286, −5.20328020249365014119067163856, −4.71196886553064058293025682932, −4.53338439786524067148189753332, −2.59468564125321312912494968264, −1.73995600075720612564854466915, −0.57698040855371956292582442167, 1.19585967816687403301798918515, 2.52020482393453193336510974908, 3.20527292750213182473308051258, 4.36265956206616400107793637084, 5.33782053400004793932224779378, 6.11435267329557235129040822386, 7.01173251253196321209836581257, 7.892450465342297622177040540556, 8.211974043092212084344041120966, 9.433972465677146110060975130851

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(2.030022195\)
\(L(\frac12)\)	\(\approx\)	\(2.030022195\)
\(L(2)\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more