LMFDB - L-function 2-72e2-8.5-c1-0-93

Dirichlet series

L(s) = 1

− 3.46i·5^-s − 3i·11^-s − 3.46i·13^-s − 3·17^-s − 7i·19^-s − 3.46·23^-s − 6.99·25^-s + 6.92i·29^-s + 6.92·31^-s − 10.3i·37^-s + 3·41^-s − 5i·43^-s − 3.46·47^-s − 7·49^-s + 13.8i·53^-s + ⋯

L(s) = 1

− 1.54i·5^-s − 0.904i·11^-s − 0.960i·13^-s − 0.727·17^-s − 1.60i·19^-s − 0.722·23^-s − 1.39·25^-s + 1.28i·29^-s + 1.24·31^-s − 1.70i·37^-s + 0.468·41^-s − 0.762i·43^-s − 0.505·47^-s − 49^-s + 1.90i·53^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$5184$ = $2^{6} \cdot 3^{4}$
Sign:	$-0.965 - 0.258i$
Analytic conductor:	$41.3944$
Root analytic conductor:	$6.43385$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{5184} (2593, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 5184,\ (\ :1/2),\ -0.965 - 0.258i)$

Particular Values

$L(1)$	$\approx$	$1.171291837$
$L(\frac12)$	$\approx$	$1.171291837$
$L(\frac{3}{2})$		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 $
good	5	$ 1 + 3.46iT - 5T^{2} $
	7	$ 1 + 7T^{2} $
	11	$ 1 + 3iT - 11T^{2} $
	13	$ 1 + 3.46iT - 13T^{2} $
	17	$ 1 + 3T + 17T^{2} $
	19	$ 1 + 7iT - 19T^{2} $
	23	$ 1 + 3.46T + 23T^{2} $
	29	$ 1 - 6.92iT - 29T^{2} $
	31	$ 1 - 6.92T + 31T^{2} $
	37	$ 1 + 10.3iT - 37T^{2} $
	41	$ 1 - 3T + 41T^{2} $
	43	$ 1 + 5iT - 43T^{2} $
	47	$ 1 + 3.46T + 47T^{2} $
	53	$ 1 - 13.8iT - 53T^{2} $
	59	$ 1 + 9iT - 59T^{2} $
	61	$ 1 - 6.92iT - 61T^{2} $
	67	$ 1 + 5iT - 67T^{2} $
	71	$ 1 - 3.46T + 71T^{2} $
	73	$ 1 + 7T + 73T^{2} $
	79	$ 1 - 17.3T + 79T^{2} $
	83	$ 1 - 12iT - 83T^{2} $
	89	$ 1 + 6T + 89T^{2} $
	97	$ 1 - T + 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

−8.060559945653913657286023926457, −7.17799572002773452705806669041, −6.25321832158480558581241662840, −5.51874370932029465239930002599, −4.91853109642496742739996272536, −4.27254591807498365442138798028, −3.27312494403157740667133781144, −2.31176670570859409219793077821, −1.06020568541231690334195133303, −0.33502219520755823074108361278, 1.71614638056039069198162432108, 2.37001415623212403603014690460, 3.29957456191312079982364560492, 4.12411258586713905597496635785, 4.77282878743524879684049644470, 6.13259008319566525409178834077, 6.39428174088352357795919330358, 7.05135491835137710731838957349, 7.87526834149681250557907727784, 8.343748665542059069673190529782

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 $\F_{q}$
Belyi maps

\(L(1)\)	\(\approx\)	\(1.171291837\)
\(L(\frac12)\)	\(\approx\)	\(1.171291837\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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