LMFDB - L-function 2-4056-13.12-c1-0-33

Dirichlet series

L(s) = 1

− 3^-s + 1.73i·5^-s − 0.732i·7^-s + 9^-s − 0.732i·11^-s − 1.73i·15^-s + 3.73·17^-s + 1.26i·19^-s + 0.732i·21^-s + 2.19·23^-s + 2.00·25^-s − 27^-s − 5.92·29^-s − 5.46i·31^-s + 0.732i·33^-s + ⋯

L(s) = 1

− 0.577·3^-s + 0.774i·5^-s − 0.276i·7^-s + 0.333·9^-s − 0.220i·11^-s − 0.447i·15^-s + 0.905·17^-s + 0.290i·19^-s + 0.159i·21^-s + 0.457·23^-s + 0.400·25^-s − 0.192·27^-s − 1.10·29^-s − 0.981i·31^-s + 0.127i·33^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 - 0.277i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.960 - 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4056$ = $2^{3} \cdot 3 \cdot 13^{2}$
Sign:	$0.960 - 0.277i$
Analytic conductor:	$32.3873$
Root analytic conductor:	$5.69098$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4056} (337, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4056,\ (\ :1/2),\ 0.960 - 0.277i)$

Particular Values

$L(1)$	$\approx$	$1.516364297$
$L(\frac12)$	$\approx$	$1.516364297$
$L(\frac{3}{2})$		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 + T $
	13	$ 1 $
good	5	$ 1 - 1.73iT - 5T^{2} $
	7	$ 1 + 0.732iT - 7T^{2} $
	11	$ 1 + 0.732iT - 11T^{2} $
	17	$ 1 - 3.73T + 17T^{2} $
	19	$ 1 - 1.26iT - 19T^{2} $
	23	$ 1 - 2.19T + 23T^{2} $
	29	$ 1 + 5.92T + 29T^{2} $
	31	$ 1 + 5.46iT - 31T^{2} $
	37	$ 1 - 8.46iT - 37T^{2} $
	41	$ 1 + 5iT - 41T^{2} $
	43	$ 1 + 1.80T + 43T^{2} $
	47	$ 1 + 6.73iT - 47T^{2} $
	53	$ 1 - 5.92T + 53T^{2} $
	59	$ 1 - 59T^{2} $
	61	$ 1 + 9.73T + 61T^{2} $
	67	$ 1 + 5.26iT - 67T^{2} $
	71	$ 1 - 8.19iT - 71T^{2} $
	73	$ 1 + 1.19iT - 73T^{2} $
	79	$ 1 - 12.3T + 79T^{2} $
	83	$ 1 + 15.6iT - 83T^{2} $
	89	$ 1 + 2.53iT - 89T^{2} $
	97	$ 1 - 10iT - 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

−8.389568903447520516497478884839, −7.51460007955168020204549155846, −7.07119292916114980473388232236, −6.22136133266763004492848220567, −5.61681419391829919009340495490, −4.79019245267763905112910036990, −3.78131961243864351261739703205, −3.12356113737960341627104580476, −1.97715606931924509452363607383, −0.74117823267867328302180158004, 0.74179462989467794669700509094, 1.70627534038878869467872676334, 2.92402514466925919784393918630, 3.94117289735092349159484857397, 4.81833225325924372773988265049, 5.38088188903049712745672033040, 6.02965099845121192360130328424, 6.99598472583678180892689646667, 7.60001542144809987231069724434, 8.466621071268487991862976445857

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 $\F_{q}$
Belyi maps

\(L(1)\)	\(\approx\)	\(1.516364297\)
\(L(\frac12)\)	\(\approx\)	\(1.516364297\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more