LMFDB - L-function 2-28e2-28.27-c3-0-33

Dirichlet series

L(s) = 1

− 7.06·3^-s + 1.22i·5^-s + 22.8·9^-s − 4.57i·11^-s − 41.0i·13^-s − 8.64i·15^-s + 119. i·17^-s − 60.1·19^-s − 8.35i·23^-s + 123.·25^-s + 29.0·27^-s + 164.·29^-s − 287.·31^-s + 32.3i·33^-s − 148.·37^-s + ⋯

L(s) = 1

− 1.35·3^-s + 0.109i·5^-s + 0.847·9^-s − 0.125i·11^-s − 0.876i·13^-s − 0.148i·15^-s + 1.70i·17^-s − 0.726·19^-s − 0.0757i·23^-s + 0.988·25^-s + 0.207·27^-s + 1.05·29^-s − 1.66·31^-s + 0.170i·33^-s − 0.660·37^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.188 + 0.981i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.188 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$784$ = $2^{4} \cdot 7^{2}$
Sign:	$-0.188 + 0.981i$
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:	$(2,\ 784,\ (\ :3/2),\ -0.188 + 0.981i)$

Particular Values

$L(2)$	$\approx$	$0.5132479976$
$L(\frac12)$	$\approx$	$0.5132479976$
$L(\frac{5}{2})$		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 + 7.06T + 27T^{2} $
	5	$ 1 - 1.22iT - 125T^{2} $
	11	$ 1 + 4.57iT - 1.33e3T^{2} $
	13	$ 1 + 41.0iT - 2.19e3T^{2} $
	17	$ 1 - 119. iT - 4.91e3T^{2} $
	19	$ 1 + 60.1T + 6.85e3T^{2} $
	23	$ 1 + 8.35iT - 1.21e4T^{2} $
	29	$ 1 - 164.T + 2.43e4T^{2} $
	31	$ 1 + 287.T + 2.97e4T^{2} $
	37	$ 1 + 148.T + 5.06e4T^{2} $
	41	$ 1 - 358. iT - 6.89e4T^{2} $
	43	$ 1 - 360. iT - 7.95e4T^{2} $
	47	$ 1 + 225.T + 1.03e5T^{2} $
	53	$ 1 - 684.T + 1.48e5T^{2} $
	59	$ 1 + 85.0T + 2.05e5T^{2} $
	61	$ 1 + 209. iT - 2.26e5T^{2} $
	67	$ 1 + 417. iT - 3.00e5T^{2} $
	71	$ 1 + 982. iT - 3.57e5T^{2} $
	73	$ 1 + 341. iT - 3.89e5T^{2} $
	79	$ 1 - 76.2iT - 4.93e5T^{2} $
	83	$ 1 - 523.T + 5.71e5T^{2} $
	89	$ 1 + 972. iT - 7.04e5T^{2} $
	97	$ 1 + 676. 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

−10.00996535319429311810353331790, −8.724444100597981174612398394086, −7.958513190119908969045378804134, −6.72046965767547296846172533726, −6.13469684714393577534448355779, −5.32509929303264077920090904823, −4.39911402373959996969054389733, −3.14911766008761980731016298280, −1.52002991233997624420613940603, −0.22098470302137037879740975861, 0.893328578198514904493793897290, 2.39344037998319745127467624856, 3.97006148348488893598303268583, 5.01283526770249933882321661190, 5.54026427559958181427023381162, 6.85774270403497173809075068647, 7.02545127912986929288380446875, 8.600825323281122561848548157386, 9.315475788458344585898460129753, 10.41130604404934556801663555063

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.5132479976\)
\(L(\frac12)\)	\(\approx\)	\(0.5132479976\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more