LMFDB - L-function 2-864-12.11-c3-0-30

Dirichlet series

L(s) = 1

+ 18.0i·5^-s − 1.63i·7^-s − 61.6·11^-s + 65.2·13^-s − 106. i·17^-s − 92.9i·19^-s − 152.·23^-s − 199.·25^-s − 235. i·29^-s + 246. i·31^-s + 29.4·35^-s + 291.·37^-s − 63.7i·41^-s − 183. i·43^-s + 229.·47^-s + ⋯

L(s) = 1

+ 1.61i·5^-s − 0.0881i·7^-s − 1.68·11^-s + 1.39·13^-s − 1.52i·17^-s − 1.12i·19^-s − 1.38·23^-s − 1.59·25^-s − 1.50i·29^-s + 1.42i·31^-s + 0.142·35^-s + 1.29·37^-s − 0.242i·41^-s − 0.649i·43^-s + 0.711·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$864$ = $2^{5} \cdot 3^{3}$
Sign:	$0.707 + 0.707i$
Analytic conductor:	$50.9776$
Root analytic conductor:	$7.13986$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{864} (863, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 864,\ (\ :3/2),\ 0.707 + 0.707i)$

Particular Values

$L(2)$	$\approx$	$1.314191565$
$L(\frac12)$	$\approx$	$1.314191565$
$L(\frac{5}{2})$		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 - 18.0iT - 125T^{2} $
	7	$ 1 + 1.63iT - 343T^{2} $
	11	$ 1 + 61.6T + 1.33e3T^{2} $
	13	$ 1 - 65.2T + 2.19e3T^{2} $
	17	$ 1 + 106. iT - 4.91e3T^{2} $
	19	$ 1 + 92.9iT - 6.85e3T^{2} $
	23	$ 1 + 152.T + 1.21e4T^{2} $
	29	$ 1 + 235. iT - 2.43e4T^{2} $
	31	$ 1 - 246. iT - 2.97e4T^{2} $
	37	$ 1 - 291.T + 5.06e4T^{2} $
	41	$ 1 + 63.7iT - 6.89e4T^{2} $
	43	$ 1 + 183. iT - 7.95e4T^{2} $
	47	$ 1 - 229.T + 1.03e5T^{2} $
	53	$ 1 - 62.5iT - 1.48e5T^{2} $
	59	$ 1 - 227.T + 2.05e5T^{2} $
	61	$ 1 - 493.T + 2.26e5T^{2} $
	67	$ 1 - 142. iT - 3.00e5T^{2} $
	71	$ 1 + 314.T + 3.57e5T^{2} $
	73	$ 1 + 758.T + 3.89e5T^{2} $
	79	$ 1 + 397. iT - 4.93e5T^{2} $
	83	$ 1 + 522.T + 5.71e5T^{2} $
	89	$ 1 - 644. iT - 7.04e5T^{2} $
	97	$ 1 - 870.T + 9.12e5T^{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

−9.990056441554836983376102169530, −8.785323050350711296392168345046, −7.75981263571158766194115455334, −7.17660479377726166805282894171, −6.25904611160990814032841460799, −5.41908885313448584993002220383, −4.11837017815985788613190808682, −2.93502952177561968995584307407, −2.40769850413766794957642619664, −0.39129585670706634579440927090, 1.01714494133092513391566132317, 2.05556009149834585624178267340, 3.70276953499747848799122099643, 4.48452273963025306480191219385, 5.73536820001200537040161105196, 5.90353108881805855582410168186, 7.75657455404233204093972829909, 8.242624311780835706281973200840, 8.796415129708014163910244807613, 9.938964028980285503716968242328

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.314191565\)
\(L(\frac12)\)	\(\approx\)	\(1.314191565\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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