LMFDB - L-function 2-6048-8.5-c1-0-52

Dirichlet series

L(s) = 1

− 1.58i·5^-s − 7^-s − 0.790i·11^-s − 0.494i·13^-s − 4.46·17^-s + 7.55i·19^-s + 0.839·23^-s + 2.48·25^-s − 2.76i·29^-s + 0.568·31^-s + 1.58i·35^-s − 0.343i·37^-s + 5.93·41^-s + 3.16i·43^-s + 4.80·47^-s + ⋯

L(s) = 1

− 0.708i·5^-s − 0.377·7^-s − 0.238i·11^-s − 0.137i·13^-s − 1.08·17^-s + 1.73i·19^-s + 0.175·23^-s + 0.497·25^-s − 0.513i·29^-s + 0.102·31^-s + 0.267i·35^-s − 0.0564i·37^-s + 0.926·41^-s + 0.482i·43^-s + 0.701·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.629 + 0.776i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.629 + 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$6048$ = $2^{5} \cdot 3^{3} \cdot 7$
Sign:	$0.629 + 0.776i$
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:	$(2,\ 6048,\ (\ :1/2),\ 0.629 + 0.776i)$

Particular Values

$L(1)$	$\approx$	$1.600663702$
$L(\frac12)$	$\approx$	$1.600663702$
$L(\frac{3}{2})$		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 + 1.58iT - 5T^{2} $
	11	$ 1 + 0.790iT - 11T^{2} $
	13	$ 1 + 0.494iT - 13T^{2} $
	17	$ 1 + 4.46T + 17T^{2} $
	19	$ 1 - 7.55iT - 19T^{2} $
	23	$ 1 - 0.839T + 23T^{2} $
	29	$ 1 + 2.76iT - 29T^{2} $
	31	$ 1 - 0.568T + 31T^{2} $
	37	$ 1 + 0.343iT - 37T^{2} $
	41	$ 1 - 5.93T + 41T^{2} $
	43	$ 1 - 3.16iT - 43T^{2} $
	47	$ 1 - 4.80T + 47T^{2} $
	53	$ 1 - 4.36iT - 53T^{2} $
	59	$ 1 + 4.50iT - 59T^{2} $
	61	$ 1 + 5.40iT - 61T^{2} $
	67	$ 1 + 7.57iT - 67T^{2} $
	71	$ 1 - 9.52T + 71T^{2} $
	73	$ 1 - 14.1T + 73T^{2} $
	79	$ 1 - 3.71T + 79T^{2} $
	83	$ 1 - 7.30iT - 83T^{2} $
	89	$ 1 + 11.5T + 89T^{2} $
	97	$ 1 - 1.75T + 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

−8.155952061646935005446364870503, −7.30403523123080901984632054070, −6.44317316881263847694428616192, −5.89093014975270527117936566679, −5.09750571463749261037443393110, −4.30463676412523097645012764755, −3.63333684981519697119432940709, −2.63152368694358757780294615407, −1.65059445942718988625445836743, −0.55545688929593120064673279021, 0.789636238363610403320788014861, 2.27636552126330435620473221295, 2.75506747842988269998559878056, 3.72636450679801537694263135281, 4.55662862589504539096554004496, 5.25117606280009473283019733804, 6.24504985044269282159730510262, 6.93237124856682055351384743551, 7.11914073658714352334878781847, 8.184900642484219340376704538996

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.600663702\)
\(L(\frac12)\)	\(\approx\)	\(1.600663702\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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