LMFDB - L-function 2-6048-8.5-c1-0-72

Dirichlet series

L(s) = 1

− 0.940i·5^-s − 7^-s + 5.98i·11^-s − 6.59i·13^-s − 2.64·17^-s + 5.83i·19^-s − 2.88·23^-s + 4.11·25^-s + 3.09i·29^-s − 3.52·31^-s + 0.940i·35^-s − 0.213i·37^-s − 1.63·41^-s − 7.16i·43^-s + 9.32·47^-s + ⋯

L(s) = 1

− 0.420i·5^-s − 0.377·7^-s + 1.80i·11^-s − 1.82i·13^-s − 0.640·17^-s + 1.33i·19^-s − 0.601·23^-s + 0.823·25^-s + 0.575i·29^-s − 0.632·31^-s + 0.158i·35^-s − 0.0351i·37^-s − 0.255·41^-s − 1.09i·43^-s + 1.36·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.600 + 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.600 + 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$6048$ = $2^{5} \cdot 3^{3} \cdot 7$
Sign:	$-0.600 + 0.799i$
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:	$(2,\ 6048,\ (\ :1/2),\ -0.600 + 0.799i)$

Particular Values

$L(1)$	$\approx$	$0.7337947644$
$L(\frac12)$	$\approx$	$0.7337947644$
$L(\frac{3}{2})$		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 + 0.940iT - 5T^{2} $
	11	$ 1 - 5.98iT - 11T^{2} $
	13	$ 1 + 6.59iT - 13T^{2} $
	17	$ 1 + 2.64T + 17T^{2} $
	19	$ 1 - 5.83iT - 19T^{2} $
	23	$ 1 + 2.88T + 23T^{2} $
	29	$ 1 - 3.09iT - 29T^{2} $
	31	$ 1 + 3.52T + 31T^{2} $
	37	$ 1 + 0.213iT - 37T^{2} $
	41	$ 1 + 1.63T + 41T^{2} $
	43	$ 1 + 7.16iT - 43T^{2} $
	47	$ 1 - 9.32T + 47T^{2} $
	53	$ 1 + 7.51iT - 53T^{2} $
	59	$ 1 + 11.9iT - 59T^{2} $
	61	$ 1 - 1.48iT - 61T^{2} $
	67	$ 1 + 13.0iT - 67T^{2} $
	71	$ 1 - 1.54T + 71T^{2} $
	73	$ 1 + 2.96T + 73T^{2} $
	79	$ 1 + 15.9T + 79T^{2} $
	83	$ 1 - 8.74iT - 83T^{2} $
	89	$ 1 - 7.50T + 89T^{2} $
	97	$ 1 - 10.7T + 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

−7.80516159529568812735307595821, −7.19596354184515089213198740674, −6.46009790278360116593259059024, −5.51365065384666571204443218904, −5.07049905184906369283746486577, −4.14335349680272156224743303049, −3.41965437253325142243833808219, −2.38602262604711252945416539762, −1.54095942128619160383034740016, −0.19878271310434394185993081777, 1.07729777590360392172838580286, 2.38689943519390139789982789313, 2.98826942310227399503666454901, 3.98415777453166174503844772429, 4.53625818901723936804232099519, 5.66447143297450662960445025050, 6.26821977740282897972971622467, 6.84150569438240520102210855824, 7.45181477067709864207595398164, 8.567842896878346323400646741928

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.7337947644\)
\(L(\frac12)\)	\(\approx\)	\(0.7337947644\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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