LMFDB - L-function 2-3024-28.27-c1-0-44

Dirichlet series

L(s) = 1

− 3.46i·5^-s + 2.64·7^-s + 4.58i·13^-s − 1.73i·17^-s + 5.29·19^-s − 4.58i·23^-s − 6.99·25^-s + 7.93·29^-s + 2.64·31^-s − 9.16i·35^-s + 4·37^-s + 6.92i·41^-s + 1.73i·43^-s + 6·47^-s + 7.00·49^-s + ⋯

L(s) = 1

− 1.54i·5^-s + 0.999·7^-s + 1.27i·13^-s − 0.420i·17^-s + 1.21·19^-s − 0.955i·23^-s − 1.39·25^-s + 1.47·29^-s + 0.475·31^-s − 1.54i·35^-s + 0.657·37^-s + 1.08i·41^-s + 0.264i·43^-s + 0.875·47^-s + 49^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3024$ = $2^{4} \cdot 3^{3} \cdot 7$
Sign:	$0.5 + 0.866i$
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:	$(2,\ 3024,\ (\ :1/2),\ 0.5 + 0.866i)$

Particular Values

$L(1)$	$\approx$	$2.235738037$
$L(\frac12)$	$\approx$	$2.235738037$
$L(\frac{3}{2})$		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 - 2.64T $
good	5	$ 1 + 3.46iT - 5T^{2} $
	11	$ 1 - 11T^{2} $
	13	$ 1 - 4.58iT - 13T^{2} $
	17	$ 1 + 1.73iT - 17T^{2} $
	19	$ 1 - 5.29T + 19T^{2} $
	23	$ 1 + 4.58iT - 23T^{2} $
	29	$ 1 - 7.93T + 29T^{2} $
	31	$ 1 - 2.64T + 31T^{2} $
	37	$ 1 - 4T + 37T^{2} $
	41	$ 1 - 6.92iT - 41T^{2} $
	43	$ 1 - 1.73iT - 43T^{2} $
	47	$ 1 - 6T + 47T^{2} $
	53	$ 1 + 7.93T + 53T^{2} $
	59	$ 1 - 3T + 59T^{2} $
	61	$ 1 - 9.16iT - 61T^{2} $
	67	$ 1 + 12.1iT - 67T^{2} $
	71	$ 1 + 4.58iT - 71T^{2} $
	73	$ 1 + 9.16iT - 73T^{2} $
	79	$ 1 - 6.92iT - 79T^{2} $
	83	$ 1 + 12T + 83T^{2} $
	89	$ 1 - 5.19iT - 89T^{2} $
	97	$ 1 + 9.16iT - 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

−8.514581560324689405446671414589, −8.097986438117028065707707305931, −7.20641934620514770326444731117, −6.25814290978141918485124794394, −5.31055211203398636729193298299, −4.56799010705388635587468425687, −4.36417634175394116819730790853, −2.81114110682084141023608228566, −1.61894045361787117320079284327, −0.871220741973297946928450288144, 1.12833511570297928311796515283, 2.45556114013393721282496942550, 3.12030548979434185576113545120, 3.98886279730996688910234514508, 5.16247616849183284838502815948, 5.76735368766186348986066713867, 6.67110633649769557792997394724, 7.47883173596284032414453816679, 7.87222200963897445084095032934, 8.700943632674904276656618446779

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.235738037\)
\(L(\frac12)\)	\(\approx\)	\(2.235738037\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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