LMFDB - L-function 2-72e2-8.5-c1-0-78

Dirichlet series

L(s) = 1

− 1.73i·5^-s + 1.26·7^-s − 1.26i·11^-s − 3i·13^-s + 4.26·17^-s − 4.19i·19^-s − 1.26·23^-s + 2.00·25^-s + 4.26i·29^-s − 3.46·31^-s − 2.19i·35^-s − 0.464i·37^-s + 3.46·41^-s − 6.19i·43^-s + 12.9·47^-s + ⋯

L(s) = 1

− 0.774i·5^-s + 0.479·7^-s − 0.382i·11^-s − 0.832i·13^-s + 1.03·17^-s − 0.962i·19^-s − 0.264·23^-s + 0.400·25^-s + 0.792i·29^-s − 0.622·31^-s − 0.371i·35^-s − 0.0762i·37^-s + 0.541·41^-s − 0.944i·43^-s + 1.88·47^-s + ⋯

Functional equation

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

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

Invariants

Degree:	$2$
Conductor:	$5184$ = $2^{6} \cdot 3^{4}$
Sign:	$-0.258 + 0.965i$
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:	$(2,\ 5184,\ (\ :1/2),\ -0.258 + 0.965i)$

Particular Values

$L(1)$	$\approx$	$1.907081034$
$L(\frac12)$	$\approx$	$1.907081034$
$L(\frac{3}{2})$		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 + 1.73iT - 5T^{2} $
	7	$ 1 - 1.26T + 7T^{2} $
	11	$ 1 + 1.26iT - 11T^{2} $
	13	$ 1 + 3iT - 13T^{2} $
	17	$ 1 - 4.26T + 17T^{2} $
	19	$ 1 + 4.19iT - 19T^{2} $
	23	$ 1 + 1.26T + 23T^{2} $
	29	$ 1 - 4.26iT - 29T^{2} $
	31	$ 1 + 3.46T + 31T^{2} $
	37	$ 1 + 0.464iT - 37T^{2} $
	41	$ 1 - 3.46T + 41T^{2} $
	43	$ 1 + 6.19iT - 43T^{2} $
	47	$ 1 - 12.9T + 47T^{2} $
	53	$ 1 - 0.928iT - 53T^{2} $
	59	$ 1 + 9.46iT - 59T^{2} $
	61	$ 1 + 6.46iT - 61T^{2} $
	67	$ 1 - 4.19iT - 67T^{2} $
	71	$ 1 + 4.73T + 71T^{2} $
	73	$ 1 - 5T + 73T^{2} $
	79	$ 1 + 14.1T + 79T^{2} $
	83	$ 1 - 10.3iT - 83T^{2} $
	89	$ 1 + 0.803T + 89T^{2} $
	97	$ 1 - 4T + 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

−8.023042333143626431834132777644, −7.43903878154449418218255833235, −6.59466697936389972469952002440, −5.44897349970160384710304239015, −5.33891028309593595284997172106, −4.36521242925281277582450480006, −3.47636940910132793093894671484, −2.60978414748067023648731594908, −1.41020536435015566580306835029, −0.54815504132620902722665549998, 1.25695659249976452179835682992, 2.18224472448400216683822737768, 3.08666090819395856337335943795, 3.98674286776720340377637047804, 4.63838765141743865266714465281, 5.70796987561597413363761177316, 6.16965660323818236743006362643, 7.20081453435997360863000850714, 7.52119839182558988243019099909, 8.341810722739898508887799260892

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.907081034\)
\(L(\frac12)\)	\(\approx\)	\(1.907081034\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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