LMFDB - L-function 2-288-8.3-c10-0-26

Dirichlet series

L(s) = 1

− 2.62e3i·5^-s + 2.25e4i·7^-s + 2.46e5·11^-s + 2.23e5i·13^-s + 2.63e6·17^-s + 1.10e6·19^-s + 7.90e6i·23^-s + 2.89e6·25^-s − 2.11e6i·29^-s − 5.09e7i·31^-s + 5.90e7·35^-s + 1.66e7i·37^-s − 1.11e8·41^-s + 6.57e7·43^-s + 1.69e8i·47^-s + ⋯

L(s) = 1

− 0.839i·5^-s + 1.33i·7^-s + 1.53·11^-s + 0.601i·13^-s + 1.85·17^-s + 0.447·19^-s + 1.22i·23^-s + 0.296·25^-s − 0.103i·29^-s − 1.77i·31^-s + 1.12·35^-s + 0.240i·37^-s − 0.961·41^-s + 0.447·43^-s + 0.739i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.730 - 0.683i)\, \overline{\Lambda}(11-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.730 - 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$288$ = $2^{5} \cdot 3^{2}$
Sign:	$0.730 - 0.683i$
Analytic conductor:	$182.982$
Root analytic conductor:	$13.5271$
Motivic weight:	$10$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{288} (271, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 288,\ (\ :5),\ 0.730 - 0.683i)$

Particular Values

$L(\frac{11}{2})$	$\approx$	$2.941760066$
$L(\frac12)$	$\approx$	$2.941760066$
$L(6)$		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 + 2.62e3iT - 9.76e6T^{2} $
	7	$ 1 - 2.25e4iT - 2.82e8T^{2} $
	11	$ 1 - 2.46e5T + 2.59e10T^{2} $
	13	$ 1 - 2.23e5iT - 1.37e11T^{2} $
	17	$ 1 - 2.63e6T + 2.01e12T^{2} $
	19	$ 1 - 1.10e6T + 6.13e12T^{2} $
	23	$ 1 - 7.90e6iT - 4.14e13T^{2} $
	29	$ 1 + 2.11e6iT - 4.20e14T^{2} $
	31	$ 1 + 5.09e7iT - 8.19e14T^{2} $
	37	$ 1 - 1.66e7iT - 4.80e15T^{2} $
	41	$ 1 + 1.11e8T + 1.34e16T^{2} $
	43	$ 1 - 6.57e7T + 2.16e16T^{2} $
	47	$ 1 - 1.69e8iT - 5.25e16T^{2} $
	53	$ 1 + 8.54e7iT - 1.74e17T^{2} $
	59	$ 1 + 3.55e8T + 5.11e17T^{2} $
	61	$ 1 - 9.81e8iT - 7.13e17T^{2} $
	67	$ 1 + 7.02e8T + 1.82e18T^{2} $
	71	$ 1 - 2.41e9iT - 3.25e18T^{2} $
	73	$ 1 + 2.80e9T + 4.29e18T^{2} $
	79	$ 1 - 1.18e9iT - 9.46e18T^{2} $
	83	$ 1 - 6.20e9T + 1.55e19T^{2} $
	89	$ 1 - 2.60e9T + 3.11e19T^{2} $
	97	$ 1 + 7.03e9T + 7.37e19T^{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

−9.764616033706243712467540069213, −9.278434155024739125563548049595, −8.487326863082824285962611712240, −7.39345465267381867727153177362, −6.04362283527909445450765277857, −5.43269930791586275154501967067, −4.23380452783987489240094028798, −3.13114559571592966045213691807, −1.72384853946861949395283342285, −0.988052904305749140574126190490, 0.66823567353792597447771636783, 1.42040614740454701485074472572, 3.14960959196841022372317164443, 3.68803545135603561303863438182, 4.94948606798201538104238119407, 6.33901623578860343528222266228, 7.02827214792827768319636038785, 7.84850814245202410085758695353, 9.089397650899158529227484381772, 10.34159905844400113170834720134

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 $\F_{q}$

\(L(\frac{11}{2})\)	\(\approx\)	\(2.941760066\)
\(L(\frac12)\)	\(\approx\)	\(2.941760066\)
\(L(6)\)		not available
\(L(1)\)		not available

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