LMFDB - L-function 2-28e2-28.27-c3-0-45

Dirichlet series

L(s) = 1

− 7·3^-s − 15.5i·5^-s + 22·9^-s + 12.1i·11^-s − 69.2i·13^-s + 109. i·15^-s − 32.9i·17^-s + 119·19^-s − 133. i·23^-s − 118·25^-s + 35·27^-s + 210·29^-s + 301·31^-s − 84.8i·33^-s − 77·37^-s + ⋯

L(s) = 1

− 1.34·3^-s − 1.39i·5^-s + 0.814·9^-s + 0.332i·11^-s − 1.47i·13^-s + 1.87i·15^-s − 0.469i·17^-s + 1.43·19^-s − 1.20i·23^-s − 0.944·25^-s + 0.249·27^-s + 1.34·29^-s + 1.74·31^-s − 0.447i·33^-s − 0.342·37^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$784$ = $2^{4} \cdot 7^{2}$
Sign:	$-0.755 + 0.654i$
Analytic conductor:	$46.2574$
Root analytic conductor:	$6.80128$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{784} (783, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 784,\ (\ :3/2),\ -0.755 + 0.654i)$

Particular Values

$L(2)$	$\approx$	$1.130754957$
$L(\frac12)$	$\approx$	$1.130754957$
$L(\frac{5}{2})$		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 $
	7	$ 1 $
good	3	$ 1 + 7T + 27T^{2} $
	5	$ 1 + 15.5iT - 125T^{2} $
	11	$ 1 - 12.1iT - 1.33e3T^{2} $
	13	$ 1 + 69.2iT - 2.19e3T^{2} $
	17	$ 1 + 32.9iT - 4.91e3T^{2} $
	19	$ 1 - 119T + 6.85e3T^{2} $
	23	$ 1 + 133. iT - 1.21e4T^{2} $
	29	$ 1 - 210T + 2.43e4T^{2} $
	31	$ 1 - 301T + 2.97e4T^{2} $
	37	$ 1 + 77T + 5.06e4T^{2} $
	41	$ 1 - 117. iT - 6.89e4T^{2} $
	43	$ 1 - 121. iT - 7.95e4T^{2} $
	47	$ 1 + 357T + 1.03e5T^{2} $
	53	$ 1 - 327T + 1.48e5T^{2} $
	59	$ 1 - 609T + 2.05e5T^{2} $
	61	$ 1 + 687. iT - 2.26e5T^{2} $
	67	$ 1 - 157. iT - 3.00e5T^{2} $
	71	$ 1 + 218. iT - 3.57e5T^{2} $
	73	$ 1 - 57.1iT - 3.89e5T^{2} $
	79	$ 1 + 812. iT - 4.93e5T^{2} $
	83	$ 1 + 588T + 5.71e5T^{2} $
	89	$ 1 + 736. iT - 7.04e5T^{2} $
	97	$ 1 - 311. iT - 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

−9.856793805662405199965788384176, −8.613951454927246511891886340641, −7.975759585656567294373869990103, −6.76098075551385359811793984362, −5.82897515418733790385485262768, −4.99912271917732708749787271509, −4.66319488744134416585174614863, −2.94775865919182988725409765903, −1.03744820017491404313415626951, −0.51776287910041232079874345725, 1.16263738638319280775398525990, 2.71018666374217934530103932717, 3.85147138565348271921374541197, 5.04531806695924021001080301759, 5.97752351647935139736548494839, 6.70185711502962838766402788443, 7.21051485179773525147386684721, 8.493826541773049244036182601605, 9.781245614838162563162547064040, 10.31260916158151608454087958680

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.130754957\)
\(L(\frac12)\)	\(\approx\)	\(1.130754957\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more