LMFDB - L-function 2-864-24.11-c3-0-8

Dirichlet series

L(s) = 1

+ 5.34·5^-s + 31.3i·7^-s + 51.3i·11^-s + 4.90i·13^-s + 76.0i·17^-s − 32.6·19^-s − 96.3·23^-s − 96.4·25^-s + 91.4·29^-s − 229. i·31^-s + 167. i·35^-s + 54.2i·37^-s − 340. i·41^-s + 416.·43^-s − 593.·47^-s + ⋯

L(s) = 1

+ 0.478·5^-s + 1.69i·7^-s + 1.40i·11^-s + 0.104i·13^-s + 1.08i·17^-s − 0.393·19^-s − 0.873·23^-s − 0.771·25^-s + 0.585·29^-s − 1.32i·31^-s + 0.810i·35^-s + 0.241i·37^-s − 1.29i·41^-s + 1.47·43^-s − 1.84·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.156i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.987 - 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$864$ = $2^{5} \cdot 3^{3}$
Sign:	$-0.987 - 0.156i$
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:	$(2,\ 864,\ (\ :3/2),\ -0.987 - 0.156i)$

Particular Values

$L(2)$	$\approx$	$1.231733025$
$L(\frac12)$	$\approx$	$1.231733025$
$L(\frac{5}{2})$		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 - 5.34T + 125T^{2} $
	7	$ 1 - 31.3iT - 343T^{2} $
	11	$ 1 - 51.3iT - 1.33e3T^{2} $
	13	$ 1 - 4.90iT - 2.19e3T^{2} $
	17	$ 1 - 76.0iT - 4.91e3T^{2} $
	19	$ 1 + 32.6T + 6.85e3T^{2} $
	23	$ 1 + 96.3T + 1.21e4T^{2} $
	29	$ 1 - 91.4T + 2.43e4T^{2} $
	31	$ 1 + 229. iT - 2.97e4T^{2} $
	37	$ 1 - 54.2iT - 5.06e4T^{2} $
	41	$ 1 + 340. iT - 6.89e4T^{2} $
	43	$ 1 - 416.T + 7.95e4T^{2} $
	47	$ 1 + 593.T + 1.03e5T^{2} $
	53	$ 1 + 229.T + 1.48e5T^{2} $
	59	$ 1 + 751. iT - 2.05e5T^{2} $
	61	$ 1 - 231. iT - 2.26e5T^{2} $
	67	$ 1 - 819.T + 3.00e5T^{2} $
	71	$ 1 - 20.8T + 3.57e5T^{2} $
	73	$ 1 - 39.6T + 3.89e5T^{2} $
	79	$ 1 + 739. iT - 4.93e5T^{2} $
	83	$ 1 - 866. iT - 5.71e5T^{2} $
	89	$ 1 - 777. iT - 7.04e5T^{2} $
	97	$ 1 - 490.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

−9.860121077022596468926338871108, −9.521393199134236118546238076311, −8.504804329597543758716361132449, −7.80046609282800653019563142031, −6.48305609420584785922550542832, −5.90211927491057169187543797438, −4.98563106694974345664523563314, −3.89693234679242746609022869077, −2.31451978265209030919362934117, −1.91784306473821058371145868465, 0.31241997427435653985245001422, 1.31704891599683544050338902564, 2.91467867113618656530377394524, 3.86658032996320061652061571768, 4.85147678010127180132754870323, 5.98773851070535202847890983587, 6.75740074873135041277326237018, 7.69214793933645616652529588419, 8.435542516437718107289668212434, 9.553306539902849392327448882491

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.231733025\)
\(L(\frac12)\)	\(\approx\)	\(1.231733025\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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