LMFDB - L-function 2-12e3-8.5-c1-0-13

Dirichlet series

L(s) = 1

− 1.73i·5^-s − 1.73·7^-s + 3i·11^-s + 4i·19^-s + 6.92·23^-s + 2.00·25^-s − 6.92i·29^-s + 1.73·31^-s + 2.99i·35^-s + 6.92i·37^-s + 12·41^-s − 4i·43^-s + 6.92·47^-s − 4·49^-s − 8.66i·53^-s + ⋯

L(s) = 1

− 0.774i·5^-s − 0.654·7^-s + 0.904i·11^-s + 0.917i·19^-s + 1.44·23^-s + 0.400·25^-s − 1.28i·29^-s + 0.311·31^-s + 0.507i·35^-s + 1.13i·37^-s + 1.87·41^-s − 0.609i·43^-s + 1.01·47^-s − 0.571·49^-s − 1.18i·53^-s + ⋯

Functional equation

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

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

Invariants

Degree:	$2$
Conductor:	$1728$ = $2^{6} \cdot 3^{3}$
Sign:	$0.965 + 0.258i$
Analytic conductor:	$13.7981$
Root analytic conductor:	$3.71458$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1728} (865, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1728,\ (\ :1/2),\ 0.965 + 0.258i)$

Particular Values

$L(1)$	$\approx$	$1.585299325$
$L(\frac12)$	$\approx$	$1.585299325$
$L(\frac{3}{2})$		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 $
good	5	$ 1 + 1.73iT - 5T^{2} $
	7	$ 1 + 1.73T + 7T^{2} $
	11	$ 1 - 3iT - 11T^{2} $
	13	$ 1 - 13T^{2} $
	17	$ 1 + 17T^{2} $
	19	$ 1 - 4iT - 19T^{2} $
	23	$ 1 - 6.92T + 23T^{2} $
	29	$ 1 + 6.92iT - 29T^{2} $
	31	$ 1 - 1.73T + 31T^{2} $
	37	$ 1 - 6.92iT - 37T^{2} $
	41	$ 1 - 12T + 41T^{2} $
	43	$ 1 + 4iT - 43T^{2} $
	47	$ 1 - 6.92T + 47T^{2} $
	53	$ 1 + 8.66iT - 53T^{2} $
	59	$ 1 - 59T^{2} $
	61	$ 1 + 13.8iT - 61T^{2} $
	67	$ 1 + 4iT - 67T^{2} $
	71	$ 1 - 13.8T + 71T^{2} $
	73	$ 1 + 7T + 73T^{2} $
	79	$ 1 + 10.3T + 79T^{2} $
	83	$ 1 - 9iT - 83T^{2} $
	89	$ 1 - 12T + 89T^{2} $
	97	$ 1 - 7T + 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

−9.421967193597420800004380587091, −8.531253322025214680518102583124, −7.75999916730005381707199925563, −6.87958882679624216702667005372, −6.09492444499203225553077473177, −5.09884466326590865152101930158, −4.40279785624555169438914778532, −3.36409475172383386513422954363, −2.19542065185260846578651047525, −0.866517614864784807904012604643, 0.891480011585764825567005429385, 2.71213234152231511023630437156, 3.13416304321371494793385260856, 4.29375335481292585763692248025, 5.40647391490685740915476853802, 6.22417144782084210461322435579, 6.98514393760380709742419250960, 7.55205633561207163351075826194, 8.908426926887122025096390991486, 9.080449595565580801015494499325

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.585299325\)
\(L(\frac12)\)	\(\approx\)	\(1.585299325\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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