LMFDB - L-function 2-8820-21.20-c1-0-15

Dirichlet series

L(s) = 1

− 5^-s − 3.51i·11^-s − 0.0229i·13^-s + 0.381·17^-s − 3.64i·19^-s + 2.90i·23^-s + 25^-s + 6.73i·29^-s + 6.02i·31^-s − 4.23·37^-s − 3.37·41^-s − 0.392·43^-s − 4.14·47^-s + 7.61i·53^-s + 3.51i·55^-s + ⋯

L(s) = 1

− 0.447·5^-s − 1.05i·11^-s − 0.00635i·13^-s + 0.0925·17^-s − 0.835i·19^-s + 0.606i·23^-s + 0.200·25^-s + 1.25i·29^-s + 1.08i·31^-s − 0.696·37^-s − 0.526·41^-s − 0.0598·43^-s − 0.605·47^-s + 1.04i·53^-s + 0.473i·55^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 - 0.698i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.716 - 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$8820$ = $2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2}$
Sign:	$0.716 - 0.698i$
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:	$(2,\ 8820,\ (\ :1/2),\ 0.716 - 0.698i)$

Particular Values

$L(1)$	$\approx$	$1.373738235$
$L(\frac12)$	$\approx$	$1.373738235$
$L(\frac{3}{2})$		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 + 3.51iT - 11T^{2} $
	13	$ 1 + 0.0229iT - 13T^{2} $
	17	$ 1 - 0.381T + 17T^{2} $
	19	$ 1 + 3.64iT - 19T^{2} $
	23	$ 1 - 2.90iT - 23T^{2} $
	29	$ 1 - 6.73iT - 29T^{2} $
	31	$ 1 - 6.02iT - 31T^{2} $
	37	$ 1 + 4.23T + 37T^{2} $
	41	$ 1 + 3.37T + 41T^{2} $
	43	$ 1 + 0.392T + 43T^{2} $
	47	$ 1 + 4.14T + 47T^{2} $
	53	$ 1 - 7.61iT - 53T^{2} $
	59	$ 1 + 1.37T + 59T^{2} $
	61	$ 1 - 1.03iT - 61T^{2} $
	67	$ 1 - 10.2T + 67T^{2} $
	71	$ 1 - 6.95iT - 71T^{2} $
	73	$ 1 + 9.99iT - 73T^{2} $
	79	$ 1 - 2.20T + 79T^{2} $
	83	$ 1 + 3.95T + 83T^{2} $
	89	$ 1 - 8.45T + 89T^{2} $
	97	$ 1 - 8.06iT - 97T^{2} $
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$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−7.88685970622112819247437080636, −7.05616923819033400594100467825, −6.64635039796060410734307181621, −5.66572767668200278043691353081, −5.13464303959277444041033103541, −4.34175616575506051525153487576, −3.33849693037038683713762102777, −3.06600740746268653438698422336, −1.76899619574109735621365308559, −0.77628782660267031131196941781, 0.41810585730735435980416051867, 1.72908400749233652273720206012, 2.43402475376195839842273029960, 3.50595767725936104755254226614, 4.13940698717850939696166133797, 4.80926242013197762601589489116, 5.57543848035035510657891508232, 6.41963489116035786821545697053, 6.96639591767600738135014665178, 7.81187623695301810221901273273

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.373738235\)
\(L(\frac12)\)	\(\approx\)	\(1.373738235\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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