LMFDB - L-function 2-6048-24.11-c1-0-59

Dirichlet series

L(s) = 1

− 0.863·5^-s + i·7^-s − 2.62i·11^-s − 1.01i·13^-s + 2.34i·17^-s + 4.09·19^-s + 6.23·23^-s − 4.25·25^-s + 1.57·29^-s − 6.29i·31^-s − 0.863i·35^-s − 5.18i·37^-s + 10.1i·41^-s + 0.496·43^-s − 5.24·47^-s + ⋯

L(s) = 1

− 0.386·5^-s + 0.377i·7^-s − 0.792i·11^-s − 0.282i·13^-s + 0.568i·17^-s + 0.939·19^-s + 1.29·23^-s − 0.850·25^-s + 0.292·29^-s − 1.13i·31^-s − 0.145i·35^-s − 0.851i·37^-s + 1.58i·41^-s + 0.0756·43^-s − 0.764·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$6048$ = $2^{5} \cdot 3^{3} \cdot 7$
Sign:	$0.642 + 0.766i$
Analytic conductor:	$48.2935$
Root analytic conductor:	$6.94935$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{6048} (5615, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 6048,\ (\ :1/2),\ 0.642 + 0.766i)$

Particular Values

$L(1)$	$\approx$	$1.605347596$
$L(\frac12)$	$\approx$	$1.605347596$
$L(\frac{3}{2})$		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 $
	7	$ 1 - iT $
good	5	$ 1 + 0.863T + 5T^{2} $
	11	$ 1 + 2.62iT - 11T^{2} $
	13	$ 1 + 1.01iT - 13T^{2} $
	17	$ 1 - 2.34iT - 17T^{2} $
	19	$ 1 - 4.09T + 19T^{2} $
	23	$ 1 - 6.23T + 23T^{2} $
	29	$ 1 - 1.57T + 29T^{2} $
	31	$ 1 + 6.29iT - 31T^{2} $
	37	$ 1 + 5.18iT - 37T^{2} $
	41	$ 1 - 10.1iT - 41T^{2} $
	43	$ 1 - 0.496T + 43T^{2} $
	47	$ 1 + 5.24T + 47T^{2} $
	53	$ 1 + 10.5T + 53T^{2} $
	59	$ 1 - 4.40iT - 59T^{2} $
	61	$ 1 + 9.59iT - 61T^{2} $
	67	$ 1 - 12.6T + 67T^{2} $
	71	$ 1 + 2.11T + 71T^{2} $
	73	$ 1 + 8.27T + 73T^{2} $
	79	$ 1 - 9.44iT - 79T^{2} $
	83	$ 1 + 6.62iT - 83T^{2} $
	89	$ 1 - 10.2iT - 89T^{2} $
	97	$ 1 - 15.1T + 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

−8.031284144281494344934005676805, −7.39700216376085619001182394574, −6.47638401749633877445699455240, −5.84743102887587689220574414377, −5.16902005460139379245107281374, −4.31475533974354139115407133305, −3.40294674950971766678901633525, −2.85235815877923769658427559439, −1.64898697303894968547134158882, −0.51768619396182043250279480428, 0.896463657554551123535157626632, 1.92707584839521682211975556428, 3.04484440967910635725045144746, 3.67706499906643058600325243628, 4.73290157049701178718404843182, 5.03521019265531245900571257746, 6.11638856388923000879974991364, 7.03794160385096781636628122614, 7.28571851678463154606676292543, 8.078242503563630571797942120406

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.605347596\)
\(L(\frac12)\)	\(\approx\)	\(1.605347596\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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