LMFDB - L-function 2-6048-24.11-c1-0-54

Dirichlet series

L(s) = 1

− 2.99·5^-s + i·7^-s + 5.36i·11^-s − 4.55i·13^-s − 0.958i·17^-s − 0.532·19^-s + 2.78·23^-s + 3.98·25^-s − 5.15·29^-s − 4.66i·31^-s − 2.99i·35^-s + 6.10i·37^-s + 12.3i·41^-s − 4.45·43^-s + 1.40·47^-s + ⋯

L(s) = 1

− 1.34·5^-s + 0.377i·7^-s + 1.61i·11^-s − 1.26i·13^-s − 0.232i·17^-s − 0.122·19^-s + 0.581·23^-s + 0.796·25^-s − 0.957·29^-s − 0.837i·31^-s − 0.506i·35^-s + 1.00i·37^-s + 1.92i·41^-s − 0.679·43^-s + 0.204·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0820 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0820 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$6048$ = $2^{5} \cdot 3^{3} \cdot 7$
Sign:	$-0.0820 + 0.996i$
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:	$(2,\ 6048,\ (\ :1/2),\ -0.0820 + 0.996i)$

Particular Values

$L(1)$	$\approx$	$0.5100782741$
$L(\frac12)$	$\approx$	$0.5100782741$
$L(\frac{3}{2})$		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 + 2.99T + 5T^{2} $
	11	$ 1 - 5.36iT - 11T^{2} $
	13	$ 1 + 4.55iT - 13T^{2} $
	17	$ 1 + 0.958iT - 17T^{2} $
	19	$ 1 + 0.532T + 19T^{2} $
	23	$ 1 - 2.78T + 23T^{2} $
	29	$ 1 + 5.15T + 29T^{2} $
	31	$ 1 + 4.66iT - 31T^{2} $
	37	$ 1 - 6.10iT - 37T^{2} $
	41	$ 1 - 12.3iT - 41T^{2} $
	43	$ 1 + 4.45T + 43T^{2} $
	47	$ 1 - 1.40T + 47T^{2} $
	53	$ 1 + 9.57T + 53T^{2} $
	59	$ 1 - 0.0356iT - 59T^{2} $
	61	$ 1 - 5.13iT - 61T^{2} $
	67	$ 1 + 8.59T + 67T^{2} $
	71	$ 1 - 13.3T + 71T^{2} $
	73	$ 1 - 6.50T + 73T^{2} $
	79	$ 1 - 1.90iT - 79T^{2} $
	83	$ 1 - 3.13iT - 83T^{2} $
	89	$ 1 + 5.91iT - 89T^{2} $
	97	$ 1 + 16.0T + 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

−7.974072490260432784361105508677, −7.29635294998711200188334368492, −6.66276835419229007024957640452, −5.66214368879602515747603418355, −4.82363197748012797492920972734, −4.35048453081909068648248070972, −3.38760761296297749152305054613, −2.71166873379877481183959677701, −1.53793029119685332514583253212, −0.17345012531038593828825871023, 0.829137205884351128877993472752, 2.08066606567912451225111729454, 3.49320125451843986545112103084, 3.59986125122271389058536814154, 4.50847813665941966317267550141, 5.34604131592650494260093295220, 6.24735463339282905198832216383, 6.95173587225709701466019058560, 7.55856732620978947103442674773, 8.251348967197320247752606976128

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.5100782741\)
\(L(\frac12)\)	\(\approx\)	\(0.5100782741\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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