LMFDB - L-function 2-3024-21.20-c1-0-62

Dirichlet series

L(s) = 1

− 0.465·5^-s + (−0.656 − 2.56i)7^-s − 5.28i·11^-s − 4.50i·13^-s + 3.15·17^-s + 1.42i·19^-s − 2.27i·23^-s − 4.78·25^-s + 6.09i·29^-s − 2.76i·31^-s + (0.305 + 1.19i)35^-s − 0.613·37^-s − 6.93·41^-s − 3.08·43^-s + 9.30·47^-s + ⋯

L(s) = 1

− 0.208·5^-s + (−0.248 − 0.968i)7^-s − 1.59i·11^-s − 1.24i·13^-s + 0.766·17^-s + 0.327i·19^-s − 0.474i·23^-s − 0.956·25^-s + 1.13i·29^-s − 0.497i·31^-s + (0.0516 + 0.201i)35^-s − 0.100·37^-s − 1.08·41^-s − 0.470·43^-s + 1.35·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.248i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3024$ = $2^{4} \cdot 3^{3} \cdot 7$
Sign:	$-0.968 + 0.248i$
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:	$(2,\ 3024,\ (\ :1/2),\ -0.968 + 0.248i)$

Particular Values

$L(1)$	$\approx$	$0.9405187825$
$L(\frac12)$	$\approx$	$0.9405187825$
$L(\frac{3}{2})$		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 + (0.656 + 2.56i)T $
good	5	$ 1 + 0.465T + 5T^{2} $
	11	$ 1 + 5.28iT - 11T^{2} $
	13	$ 1 + 4.50iT - 13T^{2} $
	17	$ 1 - 3.15T + 17T^{2} $
	19	$ 1 - 1.42iT - 19T^{2} $
	23	$ 1 + 2.27iT - 23T^{2} $
	29	$ 1 - 6.09iT - 29T^{2} $
	31	$ 1 + 2.76iT - 31T^{2} $
	37	$ 1 + 0.613T + 37T^{2} $
	41	$ 1 + 6.93T + 41T^{2} $
	43	$ 1 + 3.08T + 43T^{2} $
	47	$ 1 - 9.30T + 47T^{2} $
	53	$ 1 - 12.1iT - 53T^{2} $
	59	$ 1 - 3.62T + 59T^{2} $
	61	$ 1 + 2.27iT - 61T^{2} $
	67	$ 1 + 13.3T + 67T^{2} $
	71	$ 1 + 7.38iT - 71T^{2} $
	73	$ 1 - 10.8iT - 73T^{2} $
	79	$ 1 + 1.17T + 79T^{2} $
	83	$ 1 + 15.0T + 83T^{2} $
	89	$ 1 - 5.39T + 89T^{2} $
	97	$ 1 + 10.4iT - 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

−8.254698228020811576564859632338, −7.72865967116827746152631434465, −6.96728099623662843573716570879, −5.92363557495909475701715877215, −5.53319830501503744229042373969, −4.32525896951717468155003363136, −3.45579647865149687406232936070, −2.97392359401181303024623796546, −1.27465244600209958627632312032, −0.30408497865373321582086250110, 1.70202169322486778217128795937, 2.37138364825019027054632663739, 3.57247200912144244129256134431, 4.43189264998195892065874868810, 5.19235936843139461943206238683, 6.05561224078160979849189041057, 6.88993417370929853746762963106, 7.47847304850616191128572571153, 8.365670001315795373293301015644, 9.150800708795795794391766034194

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.9405187825\)
\(L(\frac12)\)	\(\approx\)	\(0.9405187825\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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