LMFDB - L-function 2-864-24.11-c3-0-13

Dirichlet series

L(s) = 1

+ 0.898·5^-s + 20.7i·7^-s − 0.351i·11^-s − 44.6i·13^-s + 58.0i·17^-s + 42.4·19^-s + 196.·23^-s − 124.·25^-s − 71.7·29^-s + 96.1i·31^-s + 18.6i·35^-s + 18.9i·37^-s + 322. i·41^-s + 264.·43^-s − 370.·47^-s + ⋯

L(s) = 1

+ 0.0803·5^-s + 1.11i·7^-s − 0.00964i·11^-s − 0.952i·13^-s + 0.828i·17^-s + 0.512·19^-s + 1.78·23^-s − 0.993·25^-s − 0.459·29^-s + 0.557i·31^-s + 0.0898i·35^-s + 0.0840i·37^-s + 1.22i·41^-s + 0.936·43^-s − 1.15·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.321 - 0.946i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.321 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$864$ = $2^{5} \cdot 3^{3}$
Sign:	$-0.321 - 0.946i$
Analytic conductor:	$50.9776$
Root analytic conductor:	$7.13986$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{864} (431, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 864,\ (\ :3/2),\ -0.321 - 0.946i)$

Particular Values

$L(2)$	$\approx$	$1.517740821$
$L(\frac12)$	$\approx$	$1.517740821$
$L(\frac{5}{2})$		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 - 0.898T + 125T^{2} $
	7	$ 1 - 20.7iT - 343T^{2} $
	11	$ 1 + 0.351iT - 1.33e3T^{2} $
	13	$ 1 + 44.6iT - 2.19e3T^{2} $
	17	$ 1 - 58.0iT - 4.91e3T^{2} $
	19	$ 1 - 42.4T + 6.85e3T^{2} $
	23	$ 1 - 196.T + 1.21e4T^{2} $
	29	$ 1 + 71.7T + 2.43e4T^{2} $
	31	$ 1 - 96.1iT - 2.97e4T^{2} $
	37	$ 1 - 18.9iT - 5.06e4T^{2} $
	41	$ 1 - 322. iT - 6.89e4T^{2} $
	43	$ 1 - 264.T + 7.95e4T^{2} $
	47	$ 1 + 370.T + 1.03e5T^{2} $
	53	$ 1 + 512.T + 1.48e5T^{2} $
	59	$ 1 - 240. iT - 2.05e5T^{2} $
	61	$ 1 + 526. iT - 2.26e5T^{2} $
	67	$ 1 - 156.T + 3.00e5T^{2} $
	71	$ 1 + 395.T + 3.57e5T^{2} $
	73	$ 1 + 723.T + 3.89e5T^{2} $
	79	$ 1 - 972. iT - 4.93e5T^{2} $
	83	$ 1 - 1.22e3iT - 5.71e5T^{2} $
	89	$ 1 - 1.23e3iT - 7.04e5T^{2} $
	97	$ 1 + 499.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

−9.922607876487278999606716493201, −9.191103349394063458840568407004, −8.372656097740399692794469437225, −7.61329090601762998161125613464, −6.44527772544573133392877289284, −5.63114237105132653397444503380, −4.90605287331140585518741851977, −3.47420053335344160232385104772, −2.59214436569789911882367903832, −1.27575180334315581795991541207, 0.41466298605799626316472856471, 1.64955499227113014210545226424, 3.06921433295215752397069053685, 4.12234692558578887548723549088, 4.95101681556823089233489006035, 6.10056590726898769792632275772, 7.22097235679331072392003468672, 7.46778676087565130944129574720, 8.862449649519701340682331531991, 9.493593229468059007001807285669

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.517740821\)
\(L(\frac12)\)	\(\approx\)	\(1.517740821\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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