LMFDB - L-function 2-6048-24.11-c1-0-14

Dirichlet series

L(s) = 1

+ 2.21·5^-s + i·7^-s + 2.42i·11^-s − 1.11i·13^-s − 0.701i·17^-s − 0.938·19^-s − 4.30·23^-s − 0.102·25^-s − 4.26·29^-s + 7.02i·31^-s + 2.21i·35^-s + 3.12i·37^-s − 0.157i·41^-s − 7.08·43^-s + 0.867·47^-s + ⋯

L(s) = 1

+ 0.989·5^-s + 0.377i·7^-s + 0.730i·11^-s − 0.309i·13^-s − 0.170i·17^-s − 0.215·19^-s − 0.897·23^-s − 0.0204·25^-s − 0.791·29^-s + 1.26i·31^-s + 0.374i·35^-s + 0.513i·37^-s − 0.0246i·41^-s − 1.08·43^-s + 0.126·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.727 - 0.686i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.727 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$6048$ = $2^{5} \cdot 3^{3} \cdot 7$
Sign:	$-0.727 - 0.686i$
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:	$(2,\ 6048,\ (\ :1/2),\ -0.727 - 0.686i)$

Particular Values

$L(1)$	$\approx$	$1.152514397$
$L(\frac12)$	$\approx$	$1.152514397$
$L(\frac{3}{2})$		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 - 2.21T + 5T^{2} $
	11	$ 1 - 2.42iT - 11T^{2} $
	13	$ 1 + 1.11iT - 13T^{2} $
	17	$ 1 + 0.701iT - 17T^{2} $
	19	$ 1 + 0.938T + 19T^{2} $
	23	$ 1 + 4.30T + 23T^{2} $
	29	$ 1 + 4.26T + 29T^{2} $
	31	$ 1 - 7.02iT - 31T^{2} $
	37	$ 1 - 3.12iT - 37T^{2} $
	41	$ 1 + 0.157iT - 41T^{2} $
	43	$ 1 + 7.08T + 43T^{2} $
	47	$ 1 - 0.867T + 47T^{2} $
	53	$ 1 + 3.91T + 53T^{2} $
	59	$ 1 - 7.28iT - 59T^{2} $
	61	$ 1 - 4.57iT - 61T^{2} $
	67	$ 1 + 15.1T + 67T^{2} $
	71	$ 1 - 6.93T + 71T^{2} $
	73	$ 1 - 7.02T + 73T^{2} $
	79	$ 1 + 6.65iT - 79T^{2} $
	83	$ 1 - 8.41iT - 83T^{2} $
	89	$ 1 - 7.34iT - 89T^{2} $
	97	$ 1 - 7.99T + 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

−8.351502050128772051027558450906, −7.65186586828742041711426316035, −6.82258763823484395706113826895, −6.19686622252885222117897090747, −5.48855414784291594728759683501, −4.90578166015155402402183502333, −3.96900172404923459133670294158, −2.97604807867144771524684439824, −2.12723393040136964361158182728, −1.44163977282570425512412250832, 0.26011054084795520428102990914, 1.61621265876316809580060611577, 2.26226368734462327770193587899, 3.37350586102892735928081303449, 4.08765827413360421666438170466, 4.99204083241242242046264261603, 5.91280312419149495977808627288, 6.14048838907830694891002363838, 7.05810529644761388844770796878, 7.86138620748581655292298135350

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.152514397\)
\(L(\frac12)\)	\(\approx\)	\(1.152514397\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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