LMFDB - L-function 2-6048-12.11-c1-0-46

Dirichlet series

L(s) = 1

− 1.93i·5^-s − i·7^-s − 1.13·11^-s − 3.19·13^-s + 5.93i·17^-s − 1.33i·19^-s + 2.96·23^-s + 1.26·25^-s + 1.65i·29^-s + 10.3i·31^-s − 1.93·35^-s + 0.538·37^-s − 9.51i·41^-s − 6.65i·43^-s + 11.1·47^-s + ⋯

L(s) = 1

− 0.863i·5^-s − 0.377i·7^-s − 0.342·11^-s − 0.885·13^-s + 1.43i·17^-s − 0.307i·19^-s + 0.618·23^-s + 0.253·25^-s + 0.307i·29^-s + 1.86i·31^-s − 0.326·35^-s + 0.0884·37^-s − 1.48i·41^-s − 1.01i·43^-s + 1.62·47^-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

$L(1)$	$\approx$	$1.712585449$
$L(\frac12)$	$\approx$	$1.712585449$
$L(\frac{3}{2})$		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 + 1.93iT - 5T^{2} $
	11	$ 1 + 1.13T + 11T^{2} $
	13	$ 1 + 3.19T + 13T^{2} $
	17	$ 1 - 5.93iT - 17T^{2} $
	19	$ 1 + 1.33iT - 19T^{2} $
	23	$ 1 - 2.96T + 23T^{2} $
	29	$ 1 - 1.65iT - 29T^{2} $
	31	$ 1 - 10.3iT - 31T^{2} $
	37	$ 1 - 0.538T + 37T^{2} $
	41	$ 1 + 9.51iT - 41T^{2} $
	43	$ 1 + 6.65iT - 43T^{2} $
	47	$ 1 - 11.1T + 47T^{2} $
	53	$ 1 + 1.68iT - 53T^{2} $
	59	$ 1 + 1.82T + 59T^{2} $
	61	$ 1 - 14.1T + 61T^{2} $
	67	$ 1 - 9.53iT - 67T^{2} $
	71	$ 1 - 10.9T + 71T^{2} $
	73	$ 1 - 0.319T + 73T^{2} $
	79	$ 1 - 12.9iT - 79T^{2} $
	83	$ 1 - 3.24T + 83T^{2} $
	89	$ 1 + 13.7iT - 89T^{2} $
	97	$ 1 - 2.33T + 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

−8.129094378370252215152342885373, −7.11813922648766084999234620142, −6.84011408910756652181286530331, −5.55514478846110043880877256234, −5.22411231245184600354024379273, −4.36476730115895982198340164317, −3.66708066731535395232615903725, −2.61993924522183007900263247780, −1.63084880231314225902820459323, −0.62142533459574712590356305878, 0.75798516337044596031206769850, 2.40411984852942158807685518903, 2.63263810518953924568862784577, 3.61996083807183632121696792194, 4.67738873595282304161727983463, 5.22601942421252707113918040009, 6.14933335739891774292399442046, 6.73947607080889619339750754022, 7.59038331097616361300686777456, 7.83961117716014731190941432572

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.712585449\)
\(L(\frac12)\)	\(\approx\)	\(1.712585449\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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