LMFDB - L-function 2-6048-24.11-c1-0-23

Dirichlet series

L(s) = 1

− 4.18·5^-s − i·7^-s − 1.86i·11^-s + 3.07i·13^-s − 0.504i·17^-s − 3.05·19^-s − 5.97·23^-s + 12.4·25^-s − 3.52·29^-s − 10.8i·31^-s + 4.18i·35^-s + 11.3i·37^-s + 7.26i·41^-s − 9.02·43^-s + 0.327·47^-s + ⋯

L(s) = 1

− 1.87·5^-s − 0.377i·7^-s − 0.563i·11^-s + 0.853i·13^-s − 0.122i·17^-s − 0.701·19^-s − 1.24·23^-s + 2.49·25^-s − 0.655·29^-s − 1.95i·31^-s + 0.706i·35^-s + 1.86i·37^-s + 1.13i·41^-s − 1.37·43^-s + 0.0477·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.236i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.971 - 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$6048$ = $2^{5} \cdot 3^{3} \cdot 7$
Sign:	$0.971 - 0.236i$
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:	$(2,\ 6048,\ (\ :1/2),\ 0.971 - 0.236i)$

Particular Values

$L(1)$	$\approx$	$0.6463040688$
$L(\frac12)$	$\approx$	$0.6463040688$
$L(\frac{3}{2})$		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 + 4.18T + 5T^{2} $
	11	$ 1 + 1.86iT - 11T^{2} $
	13	$ 1 - 3.07iT - 13T^{2} $
	17	$ 1 + 0.504iT - 17T^{2} $
	19	$ 1 + 3.05T + 19T^{2} $
	23	$ 1 + 5.97T + 23T^{2} $
	29	$ 1 + 3.52T + 29T^{2} $
	31	$ 1 + 10.8iT - 31T^{2} $
	37	$ 1 - 11.3iT - 37T^{2} $
	41	$ 1 - 7.26iT - 41T^{2} $
	43	$ 1 + 9.02T + 43T^{2} $
	47	$ 1 - 0.327T + 47T^{2} $
	53	$ 1 + 8.32T + 53T^{2} $
	59	$ 1 + 5.98iT - 59T^{2} $
	61	$ 1 + 13.2iT - 61T^{2} $
	67	$ 1 - 9.21T + 67T^{2} $
	71	$ 1 + 9.02T + 71T^{2} $
	73	$ 1 - 0.416T + 73T^{2} $
	79	$ 1 - 6.00iT - 79T^{2} $
	83	$ 1 - 2.62iT - 83T^{2} $
	89	$ 1 + 5.69iT - 89T^{2} $
	97	$ 1 - 6.21T + 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

−8.084453026571725704973438441654, −7.60435747937960381193072225335, −6.66378556696856088691643060492, −6.25028873837249278457128205438, −4.94076196628574089920440784822, −4.33432794309588393826907837214, −3.76992488413194078181404399594, −3.09367248627161745692625169331, −1.82485352323542995041210039960, −0.47488582817767521783999094079, 0.35594510225506735499641263918, 1.80789586458613037397770757353, 2.94834288502724505387508211147, 3.69882748620733922298566372595, 4.26625435626819665386830544563, 5.07344417345628666616278019822, 5.86201042929808080810551843760, 6.90702574078581539646564954407, 7.37977896619269695069822178633, 8.055601040076776196599511434534

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.6463040688\)
\(L(\frac12)\)	\(\approx\)	\(0.6463040688\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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