LMFDB - L-function 2-6048-12.11-c1-0-12

Dirichlet series

L(s) = 1

+ 4.29i·5^-s + i·7^-s − 4.77·11^-s − 0.0836·13^-s + 2.60i·17^-s + 4.85i·19^-s + 4.78·23^-s − 13.4·25^-s − 0.263i·29^-s + 0.304i·31^-s − 4.29·35^-s + 9.56·37^-s + 10.5i·41^-s − 4.24i·43^-s − 9.85·47^-s + ⋯

L(s) = 1

+ 1.92i·5^-s + 0.377i·7^-s − 1.43·11^-s − 0.0231·13^-s + 0.632i·17^-s + 1.11i·19^-s + 0.996·23^-s − 2.69·25^-s − 0.0488i·29^-s + 0.0546i·31^-s − 0.726·35^-s + 1.57·37^-s + 1.65i·41^-s − 0.647i·43^-s − 1.43·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$6048$ = $2^{5} \cdot 3^{3} \cdot 7$
Sign:	$-0.707 + 0.707i$
Analytic conductor:	$48.2935$
Root analytic conductor:	$6.94935$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{6048} (2591, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 6048,\ (\ :1/2),\ -0.707 + 0.707i)$

Particular Values

$L(1)$	$\approx$	$0.8549924311$
$L(\frac12)$	$\approx$	$0.8549924311$
$L(\frac{3}{2})$		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 - 4.29iT - 5T^{2} $
	11	$ 1 + 4.77T + 11T^{2} $
	13	$ 1 + 0.0836T + 13T^{2} $
	17	$ 1 - 2.60iT - 17T^{2} $
	19	$ 1 - 4.85iT - 19T^{2} $
	23	$ 1 - 4.78T + 23T^{2} $
	29	$ 1 + 0.263iT - 29T^{2} $
	31	$ 1 - 0.304iT - 31T^{2} $
	37	$ 1 - 9.56T + 37T^{2} $
	41	$ 1 - 10.5iT - 41T^{2} $
	43	$ 1 + 4.24iT - 43T^{2} $
	47	$ 1 + 9.85T + 47T^{2} $
	53	$ 1 - 11.9iT - 53T^{2} $
	59	$ 1 - 5.50T + 59T^{2} $
	61	$ 1 + 8.62T + 61T^{2} $
	67	$ 1 - 1.80iT - 67T^{2} $
	71	$ 1 + 10.8T + 71T^{2} $
	73	$ 1 + 4.88T + 73T^{2} $
	79	$ 1 + 14.3iT - 79T^{2} $
	83	$ 1 - 10.4T + 83T^{2} $
	89	$ 1 - 7.85iT - 89T^{2} $
	97	$ 1 + 1.29T + 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

−8.196818610240187970186856889753, −7.76896572091914993426998675080, −7.18903248212038713479341764241, −6.21974740075884577652106537651, −5.99508031646012975667187875609, −4.99592114848699685556704752222, −3.98596594091835624998261590244, −2.96957178842509979737541940422, −2.78280029329075592525691266367, −1.70250136152848979358903952546, 0.24310786347430728353857114589, 0.961752697557961792999523556966, 2.12874180197534926444491311027, 3.07442816261977578157041197100, 4.21176219993687116799335647940, 4.98749897418931503585900724647, 5.09430248949306407229840007823, 6.03620828287775516642692144026, 7.12972727522072825815405020839, 7.72523387124255530758134327087

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.8549924311\)
\(L(\frac12)\)	\(\approx\)	\(0.8549924311\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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