LMFDB - L-function 2-8820-21.20-c1-0-38

Dirichlet series

L(s) = 1

− 5^-s + 6.52i·11^-s − 5.03i·13^-s − 0.984·17^-s − 3.16i·19^-s + 3.84i·23^-s + 25^-s − 2.42i·29^-s + 1.31i·31^-s − 6.44·37^-s − 6.45·41^-s + 0.687·43^-s − 0.572·47^-s + 5.68i·53^-s − 6.52i·55^-s + ⋯

L(s) = 1

− 0.447·5^-s + 1.96i·11^-s − 1.39i·13^-s − 0.238·17^-s − 0.725i·19^-s + 0.802i·23^-s + 0.200·25^-s − 0.449i·29^-s + 0.235i·31^-s − 1.06·37^-s − 1.00·41^-s + 0.104·43^-s − 0.0835·47^-s + 0.780i·53^-s − 0.880i·55^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.192 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.192 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$8820$ = $2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2}$
Sign:	$0.192 + 0.981i$
Analytic conductor:	$70.4280$
Root analytic conductor:	$8.39214$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{8820} (881, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 8820,\ (\ :1/2),\ 0.192 + 0.981i)$

Particular Values

$L(1)$	$\approx$	$1.046985534$
$L(\frac12)$	$\approx$	$1.046985534$
$L(\frac{3}{2})$		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 $
	5	$ 1 + T $
	7	$ 1 $
good	11	$ 1 - 6.52iT - 11T^{2} $
	13	$ 1 + 5.03iT - 13T^{2} $
	17	$ 1 + 0.984T + 17T^{2} $
	19	$ 1 + 3.16iT - 19T^{2} $
	23	$ 1 - 3.84iT - 23T^{2} $
	29	$ 1 + 2.42iT - 29T^{2} $
	31	$ 1 - 1.31iT - 31T^{2} $
	37	$ 1 + 6.44T + 37T^{2} $
	41	$ 1 + 6.45T + 41T^{2} $
	43	$ 1 - 0.687T + 43T^{2} $
	47	$ 1 + 0.572T + 47T^{2} $
	53	$ 1 - 5.68iT - 53T^{2} $
	59	$ 1 - 8.83T + 59T^{2} $
	61	$ 1 + 11.4iT - 61T^{2} $
	67	$ 1 - 3.78T + 67T^{2} $
	71	$ 1 + 10.5iT - 71T^{2} $
	73	$ 1 + 9.92iT - 73T^{2} $
	79	$ 1 + 2.20T + 79T^{2} $
	83	$ 1 - 11.3T + 83T^{2} $
	89	$ 1 + 0.0489T + 89T^{2} $
	97	$ 1 - 13.0iT - 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

−7.68419452942632524083811565826, −6.97508709232302955799171410018, −6.41992715502653716949421837719, −5.19592703753189535540377661234, −5.02080253689016007954525769792, −4.06770894871015877108053300901, −3.33085622478120321128063196449, −2.43447364839472842288740664566, −1.58941275380998000554506380211, −0.29205130640146791847317550704, 0.858802708416759994777815169695, 1.92264408173916873869430030393, 2.94801002298503437235465923125, 3.70094509736042986211308163884, 4.22791617678510833763007697373, 5.21881427819962787381118111547, 5.84784650677659748258488107974, 6.64641524864868491060999023801, 7.04369670607733685777121828274, 8.108730792464366599197317828098

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.046985534\)
\(L(\frac12)\)	\(\approx\)	\(1.046985534\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more