LMFDB - L-function 2-2400-40.29-c1-0-9

Dirichlet series

L(s) = 1

− 3^-s + 4.68i·7^-s + 9^-s − 2.29i·11^-s + 4.97·13^-s + 2.97i·17^-s + 2.68i·19^-s − 4.68i·21^-s − 2.68i·23^-s − 27^-s − 2i·29^-s + 6.97·31^-s + 2.29i·33^-s + 4.39·37^-s − 4.97·39^-s + ⋯

L(s) = 1

− 0.577·3^-s + 1.77i·7^-s + 0.333·9^-s − 0.691i·11^-s + 1.38·13^-s + 0.722i·17^-s + 0.616i·19^-s − 1.02i·21^-s − 0.560i·23^-s − 0.192·27^-s − 0.371i·29^-s + 1.25·31^-s + 0.399i·33^-s + 0.722·37^-s − 0.797·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0310 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0310 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2400$ = $2^{5} \cdot 3 \cdot 5^{2}$
Sign:	$0.0310 - 0.999i$
Analytic conductor:	$19.1640$
Root analytic conductor:	$4.37768$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2400} (49, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2400,\ (\ :1/2),\ 0.0310 - 0.999i)$

Particular Values

$L(1)$	$\approx$	$1.418250776$
$L(\frac12)$	$\approx$	$1.418250776$
$L(\frac{3}{2})$		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 $
	5	$ 1 $
good	7	$ 1 - 4.68iT - 7T^{2} $
	11	$ 1 + 2.29iT - 11T^{2} $
	13	$ 1 - 4.97T + 13T^{2} $
	17	$ 1 - 2.97iT - 17T^{2} $
	19	$ 1 - 2.68iT - 19T^{2} $
	23	$ 1 + 2.68iT - 23T^{2} $
	29	$ 1 + 2iT - 29T^{2} $
	31	$ 1 - 6.97T + 31T^{2} $
	37	$ 1 - 4.39T + 37T^{2} $
	41	$ 1 + 11.3T + 41T^{2} $
	43	$ 1 - 9.37T + 43T^{2} $
	47	$ 1 - 7.27iT - 47T^{2} $
	53	$ 1 - 2T + 53T^{2} $
	59	$ 1 - 1.70iT - 59T^{2} $
	61	$ 1 - 4.58iT - 61T^{2} $
	67	$ 1 + 4T + 67T^{2} $
	71	$ 1 + 0.585T + 71T^{2} $
	73	$ 1 + 6iT - 73T^{2} $
	79	$ 1 - 1.02T + 79T^{2} $
	83	$ 1 + 13.3T + 83T^{2} $
	89	$ 1 + 3.37T + 89T^{2} $
	97	$ 1 - 3.95iT - 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

−8.893948204729404010132886326067, −8.554256138720850927108114534234, −7.81714457236639567265712233824, −6.30493582344699550600534132051, −6.15482679441086505568128651586, −5.47464514385573775393390803243, −4.43271736156641206076866062438, −3.39323298630402956298264283835, −2.40317585666911572209884099399, −1.21391242591165334624701638485, 0.60533226567943684770470271457, 1.53049195324949182375990917473, 3.14071082654360387145378940391, 4.09309291493534630291159479639, 4.64050388347581146855068942298, 5.64175729498501102887656793991, 6.70631863133614594610455683004, 7.04057120406156320249661298158, 7.85784637966337449074587605655, 8.747373732451683301699287631232

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.418250776\)
\(L(\frac12)\)	\(\approx\)	\(1.418250776\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more