LMFDB - L-function 2-3024-21.20-c1-0-30

Dirichlet series

L(s) = 1

− 1.73·5^-s + (−2 + 1.73i)7^-s + 3i·11^-s − 3.46i·13^-s − 6.92·17^-s + 1.73i·19^-s + 3i·23^-s − 2.00·25^-s − 6i·29^-s − 5.19i·31^-s + (3.46 − 2.99i)35^-s + 7·37^-s + 12.1·41^-s + 2·43^-s − 3.46·47^-s + ⋯

L(s) = 1

− 0.774·5^-s + (−0.755 + 0.654i)7^-s + 0.904i·11^-s − 0.960i·13^-s − 1.68·17^-s + 0.397i·19^-s + 0.625i·23^-s − 0.400·25^-s − 1.11i·29^-s − 0.933i·31^-s + (0.585 − 0.507i)35^-s + 1.15·37^-s + 1.89·41^-s + 0.304·43^-s − 0.505·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 + 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3024$ = $2^{4} \cdot 3^{3} \cdot 7$
Sign:	$0.654 + 0.755i$
Analytic conductor:	$24.1467$
Root analytic conductor:	$4.91393$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3024} (1889, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3024,\ (\ :1/2),\ 0.654 + 0.755i)$

Particular Values

$L(1)$	$\approx$	$0.8206606186$
$L(\frac12)$	$\approx$	$0.8206606186$
$L(\frac{3}{2})$		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 $
	7	$ 1 + (2 - 1.73i)T $
good	5	$ 1 + 1.73T + 5T^{2} $
	11	$ 1 - 3iT - 11T^{2} $
	13	$ 1 + 3.46iT - 13T^{2} $
	17	$ 1 + 6.92T + 17T^{2} $
	19	$ 1 - 1.73iT - 19T^{2} $
	23	$ 1 - 3iT - 23T^{2} $
	29	$ 1 + 6iT - 29T^{2} $
	31	$ 1 + 5.19iT - 31T^{2} $
	37	$ 1 - 7T + 37T^{2} $
	41	$ 1 - 12.1T + 41T^{2} $
	43	$ 1 - 2T + 43T^{2} $
	47	$ 1 + 3.46T + 47T^{2} $
	53	$ 1 - 12iT - 53T^{2} $
	59	$ 1 - 3.46T + 59T^{2} $
	61	$ 1 + 6.92iT - 61T^{2} $
	67	$ 1 + 2T + 67T^{2} $
	71	$ 1 + 3iT - 71T^{2} $
	73	$ 1 - 3.46iT - 73T^{2} $
	79	$ 1 - 10T + 79T^{2} $
	83	$ 1 + 17.3T + 83T^{2} $
	89	$ 1 - 5.19T + 89T^{2} $
	97	$ 1 + 13.8iT - 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

−8.579757065556454454269860115163, −7.78420947162146696095936967058, −7.29853292313131711800958419318, −6.21467530392532564051532886450, −5.76600076727854528197094724023, −4.51353866304988817401188276589, −4.01514851054813665090003390018, −2.87887968375824336203322480100, −2.12393609344801081837188163161, −0.36103029811922878218604411531, 0.78166000136720687015148149888, 2.32852795461321131309851571246, 3.34179720937903262778843681848, 4.12612018602031606189479632411, 4.69268245711063308092133663035, 5.97999394463375606826302926304, 6.69967744420730424624621056236, 7.16210681859465324956638943809, 8.126904193692300967873193914657, 8.870124421908348015838614007022

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.8206606186\)
\(L(\frac12)\)	\(\approx\)	\(0.8206606186\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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