LMFDB - L-function 2-6048-8.5-c1-0-35

Dirichlet series

L(s) = 1

+ 0.356i·5^-s + 7^-s + 5.96i·11^-s + 2.87i·13^-s + 3.16·17^-s − 0.127i·19^-s + 2.80·23^-s + 4.87·25^-s − 2.44i·29^-s + 7·31^-s + 0.356i·35^-s − 1.87i·37^-s − 6.99·41^-s − 1.12i·43^-s + 7.34·47^-s + ⋯

L(s) = 1

+ 0.159i·5^-s + 0.377·7^-s + 1.79i·11^-s + 0.796i·13^-s + 0.766·17^-s − 0.0291i·19^-s + 0.585·23^-s + 0.974·25^-s − 0.454i·29^-s + 1.25·31^-s + 0.0602i·35^-s − 0.307i·37^-s − 1.09·41^-s − 0.171i·43^-s + 1.07·47^-s + ⋯

Functional equation

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

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

Invariants

Degree:	$2$
Conductor:	$6048$ = $2^{5} \cdot 3^{3} \cdot 7$
Sign:	$0.126 - 0.992i$
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:	$(2,\ 6048,\ (\ :1/2),\ 0.126 - 0.992i)$

Particular Values

$L(1)$	$\approx$	$2.110672748$
$L(\frac12)$	$\approx$	$2.110672748$
$L(\frac{3}{2})$		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 - 0.356iT - 5T^{2} $
	11	$ 1 - 5.96iT - 11T^{2} $
	13	$ 1 - 2.87iT - 13T^{2} $
	17	$ 1 - 3.16T + 17T^{2} $
	19	$ 1 + 0.127iT - 19T^{2} $
	23	$ 1 - 2.80T + 23T^{2} $
	29	$ 1 + 2.44iT - 29T^{2} $
	31	$ 1 - 7T + 31T^{2} $
	37	$ 1 + 1.87iT - 37T^{2} $
	41	$ 1 + 6.99T + 41T^{2} $
	43	$ 1 + 1.12iT - 43T^{2} $
	47	$ 1 - 7.34T + 47T^{2} $
	53	$ 1 - 53T^{2} $
	59	$ 1 - 6.63iT - 59T^{2} $
	61	$ 1 + 13.7iT - 61T^{2} $
	67	$ 1 - 8.61iT - 67T^{2} $
	71	$ 1 + 5.96T + 71T^{2} $
	73	$ 1 - 0.872T + 73T^{2} $
	79	$ 1 + 12.6T + 79T^{2} $
	83	$ 1 - 6.63iT - 83T^{2} $
	89	$ 1 + 6.99T + 89T^{2} $
	97	$ 1 + 9.74T + 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

−8.227345920436781027120124634924, −7.31019115252661511005576110907, −7.03730888196519360834995343749, −6.21168451412933263501968503588, −5.19717980471381120765343763932, −4.64116307806230046589580248023, −3.99155428195168967774439047280, −2.86609725201639784332249962472, −2.06173850842122395883349815189, −1.15590604749063937512279275809, 0.61590984346459155109150949118, 1.36825121604899471126493705249, 2.93373838658257627966009446600, 3.16567433713563689814407166149, 4.28904205375763315420395457526, 5.19058478522042885089705382405, 5.68000943845942255321817875039, 6.41505890689860611700383018600, 7.27048785628985590470788598739, 8.068286815411648599197162787173

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.110672748\)
\(L(\frac12)\)	\(\approx\)	\(2.110672748\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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