LMFDB - L-function 2-3024-28.27-c1-0-11

Dirichlet series

L(s) = 1

− 3.63i·5^-s + (−1 − 2.44i)7^-s + 1.50i·11^-s + 5.91i·13^-s + 5.14i·17^-s + 5.24·19^-s + 8.78i·23^-s − 8.24·25^-s − 8.91·29^-s + 3.24·31^-s + (−8.91 + 3.63i)35^-s + 37^-s − 6.65i·41^-s + 9.37i·43^-s − 3.69·47^-s + ⋯

L(s) = 1

− 1.62i·5^-s + (−0.377 − 0.925i)7^-s + 0.454i·11^-s + 1.64i·13^-s + 1.24i·17^-s + 1.20·19^-s + 1.83i·23^-s − 1.64·25^-s − 1.65·29^-s + 0.582·31^-s + (−1.50 + 0.615i)35^-s + 0.164·37^-s − 1.03i·41^-s + 1.43i·43^-s − 0.538·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.612 - 0.790i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3024$ = $2^{4} \cdot 3^{3} \cdot 7$
Sign:	$0.612 - 0.790i$
Analytic conductor:	$24.1467$
Root analytic conductor:	$4.91393$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3024} (1567, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3024,\ (\ :1/2),\ 0.612 - 0.790i)$

Particular Values

$L(1)$	$\approx$	$1.175072133$
$L(\frac12)$	$\approx$	$1.175072133$
$L(\frac{3}{2})$		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 + (1 + 2.44i)T $
good	5	$ 1 + 3.63iT - 5T^{2} $
	11	$ 1 - 1.50iT - 11T^{2} $
	13	$ 1 - 5.91iT - 13T^{2} $
	17	$ 1 - 5.14iT - 17T^{2} $
	19	$ 1 - 5.24T + 19T^{2} $
	23	$ 1 - 8.78iT - 23T^{2} $
	29	$ 1 + 8.91T + 29T^{2} $
	31	$ 1 - 3.24T + 31T^{2} $
	37	$ 1 - T + 37T^{2} $
	41	$ 1 + 6.65iT - 41T^{2} $
	43	$ 1 - 9.37iT - 43T^{2} $
	47	$ 1 + 3.69T + 47T^{2} $
	53	$ 1 + 12.6T + 53T^{2} $
	59	$ 1 - 8.91T + 59T^{2} $
	61	$ 1 + 3.46iT - 61T^{2} $
	67	$ 1 - 10.8iT - 67T^{2} $
	71	$ 1 + 3.63iT - 71T^{2} $
	73	$ 1 - 0.420iT - 73T^{2} $
	79	$ 1 - 4.47iT - 79T^{2} $
	83	$ 1 - 3.69T + 83T^{2} $
	89	$ 1 - 13.9iT - 89T^{2} $
	97	$ 1 - 13.2iT - 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

−9.016770371524814724854268285829, −8.002744995003767400774947916603, −7.48119738664847095318448569638, −6.60007928249526414460367139641, −5.66507514524857527642123695669, −4.90629174775449708332411557684, −4.10702539648904277797326660461, −3.59933753312088439606553740594, −1.78134620773861925201347113320, −1.22847883729242143737330354093, 0.38546299892492679391648746798, 2.30879275957575001832926948158, 3.05147224486631224189786822523, 3.34970779435137782981383851708, 4.91671958664972324886807819847, 5.72044152806688230892050144511, 6.30158129534093397336755970914, 7.12208461161871471077660681982, 7.75567278094872124373583087686, 8.546390835016822739831258240354

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.175072133\)
\(L(\frac12)\)	\(\approx\)	\(1.175072133\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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