LMFDB - L-function 2-12e3-8.3-c2-0-6

Dirichlet series

L(s) = 1

− 8.36i·5^-s − 2.43i·7^-s + 2.41·11^-s + 20.8i·13^-s + 13.1·17^-s − 32.7·19^-s + 33.8i·23^-s − 45.0·25^-s + 34.7i·29^-s − 33.7i·31^-s − 20.4·35^-s + 30.8i·37^-s − 35.1·41^-s − 27.9·43^-s − 21.9i·47^-s + ⋯

L(s) = 1

− 1.67i·5^-s − 0.348i·7^-s + 0.219·11^-s + 1.60i·13^-s + 0.775·17^-s − 1.72·19^-s + 1.47i·23^-s − 1.80·25^-s + 1.19i·29^-s − 1.08i·31^-s − 0.583·35^-s + 0.833i·37^-s − 0.857·41^-s − 0.649·43^-s − 0.467i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1728$ = $2^{6} \cdot 3^{3}$
Sign:	$0.258 - 0.965i$
Analytic conductor:	$47.0845$
Root analytic conductor:	$6.86182$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1728} (1567, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1728,\ (\ :1),\ 0.258 - 0.965i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.9631988318$
$L(\frac12)$	$\approx$	$0.9631988318$
$L(2)$		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 + 8.36iT - 25T^{2} $
	7	$ 1 + 2.43iT - 49T^{2} $
	11	$ 1 - 2.41T + 121T^{2} $
	13	$ 1 - 20.8iT - 169T^{2} $
	17	$ 1 - 13.1T + 289T^{2} $
	19	$ 1 + 32.7T + 361T^{2} $
	23	$ 1 - 33.8iT - 529T^{2} $
	29	$ 1 - 34.7iT - 841T^{2} $
	31	$ 1 + 33.7iT - 961T^{2} $
	37	$ 1 - 30.8iT - 1.36e3T^{2} $
	41	$ 1 + 35.1T + 1.68e3T^{2} $
	43	$ 1 + 27.9T + 1.84e3T^{2} $
	47	$ 1 + 21.9iT - 2.20e3T^{2} $
	53	$ 1 - 26.3iT - 2.80e3T^{2} $
	59	$ 1 + 93.3T + 3.48e3T^{2} $
	61	$ 1 - 27.7iT - 3.72e3T^{2} $
	67	$ 1 - 9.97T + 4.48e3T^{2} $
	71	$ 1 - 64.5iT - 5.04e3T^{2} $
	73	$ 1 + 45.5T + 5.32e3T^{2} $
	79	$ 1 - 82.8iT - 6.24e3T^{2} $
	83	$ 1 - 59.4T + 6.88e3T^{2} $
	89	$ 1 - 139.T + 7.92e3T^{2} $
	97	$ 1 - 150.T + 9.40e3T^{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

−9.160509405496878751553171030089, −8.674189561781164605775319868308, −7.86287626869398274944239956375, −6.92391929428639707941010391965, −6.01594417687306759035234447420, −5.07172441740383414251477613497, −4.37848446444999420562504086044, −3.67542621866137033918581786631, −1.92298122522672124003944816769, −1.20066916941258965829655199876, 0.25955187183058172600715307462, 2.12961066210105318447015868143, 2.92317105238350329468982519310, 3.66449088445343843161732499207, 4.88124575559804164820176111241, 6.10413628011080429667107325236, 6.36348975388416711075167325065, 7.41098588188314147012739303434, 8.076857200590514921127722398165, 8.882192290438118268518803923826

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.9631988318\)
\(L(\frac12)\)	\(\approx\)	\(0.9631988318\)
\(L(2)\)		not available
\(L(1)\)		not available

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