LMFDB - L-function 2-2880-8.5-c1-0-12

Dirichlet series

L(s) = 1

− i·5^-s − 4.73·7^-s + 3.46i·11^-s − 3.46i·13^-s + 3.46·17^-s + 2i·19^-s − 2.19·23^-s − 25^-s + 2.53·31^-s + 4.73i·35^-s − 6i·37^-s − 9.46·41^-s + 0.196i·43^-s + 2.19·47^-s + 15.3·49^-s + ⋯

L(s) = 1

− 0.447i·5^-s − 1.78·7^-s + 1.04i·11^-s − 0.960i·13^-s + 0.840·17^-s + 0.458i·19^-s − 0.457·23^-s − 0.200·25^-s + 0.455·31^-s + 0.799i·35^-s − 0.986i·37^-s − 1.47·41^-s + 0.0299i·43^-s + 0.320·47^-s + 2.19·49^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2880$ = $2^{6} \cdot 3^{2} \cdot 5$
Sign:	$0.965 - 0.258i$
Analytic conductor:	$22.9969$
Root analytic conductor:	$4.79550$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2880} (1441, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2880,\ (\ :1/2),\ 0.965 - 0.258i)$

Particular Values

$L(1)$	$\approx$	$1.196137083$
$L(\frac12)$	$\approx$	$1.196137083$
$L(\frac{3}{2})$		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 $
	5	$ 1 + iT $
good	7	$ 1 + 4.73T + 7T^{2} $
	11	$ 1 - 3.46iT - 11T^{2} $
	13	$ 1 + 3.46iT - 13T^{2} $
	17	$ 1 - 3.46T + 17T^{2} $
	19	$ 1 - 2iT - 19T^{2} $
	23	$ 1 + 2.19T + 23T^{2} $
	29	$ 1 - 29T^{2} $
	31	$ 1 - 2.53T + 31T^{2} $
	37	$ 1 + 6iT - 37T^{2} $
	41	$ 1 + 9.46T + 41T^{2} $
	43	$ 1 - 0.196iT - 43T^{2} $
	47	$ 1 - 2.19T + 47T^{2} $
	53	$ 1 - 10.3iT - 53T^{2} $
	59	$ 1 - 6iT - 59T^{2} $
	61	$ 1 - 0.928iT - 61T^{2} $
	67	$ 1 - 0.196iT - 67T^{2} $
	71	$ 1 - 16.3T + 71T^{2} $
	73	$ 1 - 6.39T + 73T^{2} $
	79	$ 1 - 12T + 79T^{2} $
	83	$ 1 - 1.26iT - 83T^{2} $
	89	$ 1 - 12.9T + 89T^{2} $
	97	$ 1 - 14.3T + 97T^{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

−8.951023034081057779880376199162, −7.958466995012848590628560503897, −7.33421259082031521011120843892, −6.46312270284939233053743924455, −5.80909311151904962056639811651, −5.00945240270312655905294740940, −3.88471281545415381491634780591, −3.25663737102043192800435737279, −2.21837508841751435191391311767, −0.74454973611065485979237099178, 0.57066619521290812419596810377, 2.20654698750174407324397915462, 3.39720505041604725841037156195, 3.49815388047164014256668781848, 4.89300288987334518846189442302, 5.92956748791203245847954121089, 6.51664311641030939405960275092, 6.95077267535885762490894814456, 8.048659817515079680875894077004, 8.790614040685718809384400785718

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.196137083\)
\(L(\frac12)\)	\(\approx\)	\(1.196137083\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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