LMFDB - L-function 2-432-12.11-c1-0-4

Dirichlet series

L(s) = 1

− 1.73i·7^-s + 7·13^-s − 8.66i·19^-s + 5·25^-s + 10.3i·31^-s − 37^-s − 10.3i·43^-s + 4·49^-s − 13·61^-s + 12.1i·67^-s − 17·73^-s + 12.1i·79^-s − 12.1i·91^-s − 5·97^-s + 19.0i·103^-s + ⋯

L(s) = 1

− 0.654i·7^-s + 1.94·13^-s − 1.98i·19^-s + 25^-s + 1.86i·31^-s − 0.164·37^-s − 1.58i·43^-s + 0.571·49^-s − 1.66·61^-s + 1.48i·67^-s − 1.98·73^-s + 1.36i·79^-s − 1.27i·91^-s − 0.507·97^-s + 1.87i·103^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$432$ = $2^{4} \cdot 3^{3}$
Sign:	$0.866 + 0.5i$
Analytic conductor:	$3.44953$
Root analytic conductor:	$1.85729$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{432} (431, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 432,\ (\ :1/2),\ 0.866 + 0.5i)$

Particular Values

$L(1)$	$\approx$	$1.41476 - 0.379085i$
$L(\frac12)$	$\approx$	$1.41476 - 0.379085i$
$L(\frac{3}{2})$		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 $
good	5	$ 1 - 5T^{2} $
	7	$ 1 + 1.73iT - 7T^{2} $
	11	$ 1 + 11T^{2} $
	13	$ 1 - 7T + 13T^{2} $
	17	$ 1 - 17T^{2} $
	19	$ 1 + 8.66iT - 19T^{2} $
	23	$ 1 + 23T^{2} $
	29	$ 1 - 29T^{2} $
	31	$ 1 - 10.3iT - 31T^{2} $
	37	$ 1 + T + 37T^{2} $
	41	$ 1 - 41T^{2} $
	43	$ 1 + 10.3iT - 43T^{2} $
	47	$ 1 + 47T^{2} $
	53	$ 1 - 53T^{2} $
	59	$ 1 + 59T^{2} $
	61	$ 1 + 13T + 61T^{2} $
	67	$ 1 - 12.1iT - 67T^{2} $
	71	$ 1 + 71T^{2} $
	73	$ 1 + 17T + 73T^{2} $
	79	$ 1 - 12.1iT - 79T^{2} $
	83	$ 1 + 83T^{2} $
	89	$ 1 - 89T^{2} $
	97	$ 1 + 5T + 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

−10.85264818012528431013622729626, −10.52233958930717249313894169368, −9.005275133993387819880779534966, −8.599536346141706453549999928472, −7.20217671530449479560334293127, −6.55213855881737620069911805785, −5.27988116682589768936923727974, −4.14149037091446659297479417917, −3.01760839698715801962750927644, −1.13293753812297903734402838270, 1.55995293248542508477899289659, 3.19359405981306094723759491957, 4.27641933862465300523732268190, 5.80810051314631591306814952085, 6.21899907388623648826181652843, 7.74239137126834312935054998100, 8.490842121873430371566472986357, 9.338906886207236034516955166648, 10.41623439274679055973854166688, 11.21201751875047913058671935927

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.41476 - 0.379085i\)
\(L(\frac12)\)	\(\approx\)	\(1.41476 - 0.379085i\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more