LMFDB - L-function 2-288-8.5-c3-0-0

Dirichlet series

L(s) = 1

− 6.32i·5^-s − 10·7^-s + 37.9i·11^-s − 59.3i·13^-s − 75.0·17^-s + 118. i·19^-s − 150.·23^-s + 85·25^-s + 246. i·29^-s − 62·31^-s + 63.2i·35^-s − 59.3i·37^-s − 375.·41^-s − 118. i·43^-s − 450.·47^-s + ⋯

L(s) = 1

− 0.565i·5^-s − 0.539·7^-s + 1.04i·11^-s − 1.26i·13^-s − 1.07·17^-s + 1.43i·19^-s − 1.36·23^-s + 0.680·25^-s + 1.57i·29^-s − 0.359·31^-s + 0.305i·35^-s − 0.263i·37^-s − 1.42·41^-s − 0.420i·43^-s − 1.39·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.829 - 0.559i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.829 - 0.559i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$288$ = $2^{5} \cdot 3^{2}$
Sign:	$-0.829 - 0.559i$
Analytic conductor:	$16.9925$
Root analytic conductor:	$4.12220$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{288} (145, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 288,\ (\ :3/2),\ -0.829 - 0.559i)$

Particular Values

$L(2)$	$\approx$	$0.3267886944$
$L(\frac12)$	$\approx$	$0.3267886944$
$L(\frac{5}{2})$		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 + 6.32iT - 125T^{2} $
	7	$ 1 + 10T + 343T^{2} $
	11	$ 1 - 37.9iT - 1.33e3T^{2} $
	13	$ 1 + 59.3iT - 2.19e3T^{2} $
	17	$ 1 + 75.0T + 4.91e3T^{2} $
	19	$ 1 - 118. iT - 6.85e3T^{2} $
	23	$ 1 + 150.T + 1.21e4T^{2} $
	29	$ 1 - 246. iT - 2.43e4T^{2} $
	31	$ 1 + 62T + 2.97e4T^{2} $
	37	$ 1 + 59.3iT - 5.06e4T^{2} $
	41	$ 1 + 375.T + 6.89e4T^{2} $
	43	$ 1 + 118. iT - 7.95e4T^{2} $
	47	$ 1 + 450.T + 1.03e5T^{2} $
	53	$ 1 - 132. iT - 1.48e5T^{2} $
	59	$ 1 - 733. iT - 2.05e5T^{2} $
	61	$ 1 - 533. iT - 2.26e5T^{2} $
	67	$ 1 + 711. iT - 3.00e5T^{2} $
	71	$ 1 + 3.57e5T^{2} $
	73	$ 1 - 30T + 3.89e5T^{2} $
	79	$ 1 + 94T + 4.93e5T^{2} $
	83	$ 1 + 670. iT - 5.71e5T^{2} $
	89	$ 1 - 750.T + 7.04e5T^{2} $
	97	$ 1 - 130T + 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

−12.03052077225610577370585825443, −10.57792832053785581881922710506, −9.984453758641049395333518744955, −8.889061978815088400324486817702, −7.973105939005372437240550521558, −6.87359066905063971362898920330, −5.71787271307614744210991680586, −4.63733058652809752898203247451, −3.35143552432572228030071450784, −1.72726583032781484984080730914, 0.11598757161119102119435961137, 2.22932242136234250296507859662, 3.49911373794932381001709001421, 4.74174444524037927031743085510, 6.35275359434611735562757618653, 6.73018309164840051319366687763, 8.182955577806507176805367752384, 9.132631268427274817199033464170, 10.01297826558520327542894029339, 11.25371417326831635815340307304

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 $\F_{q}$
Belyi maps

\(L(2)\)	\(\approx\)	\(0.3267886944\)
\(L(\frac12)\)	\(\approx\)	\(0.3267886944\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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