LMFDB - L-function 2-6048-8.5-c1-0-93

Dirichlet series

L(s) = 1

− 2.80i·5^-s + 7^-s − 3.51i·11^-s − 4.87i·13^-s − 3.16·17^-s − 7.87i·19^-s − 0.356·23^-s − 2.87·25^-s − 2.44i·29^-s + 7·31^-s − 2.80i·35^-s + 5.87i·37^-s − 10.1·41^-s − 8.87i·43^-s + 7.34·47^-s + ⋯

L(s) = 1

− 1.25i·5^-s + 0.377·7^-s − 1.06i·11^-s − 1.35i·13^-s − 0.766·17^-s − 1.80i·19^-s − 0.0743·23^-s − 0.574·25^-s − 0.454i·29^-s + 1.25·31^-s − 0.474i·35^-s + 0.965i·37^-s − 1.58·41^-s − 1.35i·43^-s + 1.07·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$6048$ = $2^{5} \cdot 3^{3} \cdot 7$
Sign:	$-0.992 + 0.126i$
Analytic conductor:	$48.2935$
Root analytic conductor:	$6.94935$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{6048} (3025, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 6048,\ (\ :1/2),\ -0.992 + 0.126i)$

Particular Values

$L(1)$	$\approx$	$1.589955074$
$L(\frac12)$	$\approx$	$1.589955074$
$L(\frac{3}{2})$		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 - T $
good	5	$ 1 + 2.80iT - 5T^{2} $
	11	$ 1 + 3.51iT - 11T^{2} $
	13	$ 1 + 4.87iT - 13T^{2} $
	17	$ 1 + 3.16T + 17T^{2} $
	19	$ 1 + 7.87iT - 19T^{2} $
	23	$ 1 + 0.356T + 23T^{2} $
	29	$ 1 + 2.44iT - 29T^{2} $
	31	$ 1 - 7T + 31T^{2} $
	37	$ 1 - 5.87iT - 37T^{2} $
	41	$ 1 + 10.1T + 41T^{2} $
	43	$ 1 + 8.87iT - 43T^{2} $
	47	$ 1 - 7.34T + 47T^{2} $
	53	$ 1 - 53T^{2} $
	59	$ 1 - 12.9iT - 59T^{2} $
	61	$ 1 - 1.74iT - 61T^{2} $
	67	$ 1 + 14.6iT - 67T^{2} $
	71	$ 1 - 3.51T + 71T^{2} $
	73	$ 1 + 6.87T + 73T^{2} $
	79	$ 1 - 10.6T + 79T^{2} $
	83	$ 1 - 12.9iT - 83T^{2} $
	89	$ 1 + 10.1T + 89T^{2} $
	97	$ 1 - 5.74T + 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

−7.991756390307830967279281496005, −7.03998515370609450765097030100, −6.24447604026180361479925936741, −5.38179441534484853721411169291, −4.95532292502069662548544854377, −4.23990342992093518922384407319, −3.18046102589403938302221360822, −2.39193587385497847038686418184, −1.04841944900664709962520175307, −0.44299111205426556689427827113, 1.63483635829392018673831712996, 2.18456319277669643861572980042, 3.18733018546493927227554408167, 4.09715004172030618775837298295, 4.61689691397757627125328483347, 5.67778189531708109825241818949, 6.55896818060072703568632816606, 6.85872774649233280461315301100, 7.60490700910351985004483941253, 8.306546136485115901752285359463

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.589955074\)
\(L(\frac12)\)	\(\approx\)	\(1.589955074\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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