LMFDB - L-function 2-6048-24.11-c1-0-56

Dirichlet series

L(s) = 1

− 0.206·5^-s − i·7^-s − 6.42i·11^-s + 3.52i·13^-s + 3.90i·17^-s + 5.48·19^-s − 0.880·23^-s − 4.95·25^-s + 3.20·29^-s − 0.0631i·31^-s + 0.206i·35^-s + 9.91i·37^-s − 5.94i·41^-s + 1.13·43^-s + 6.81·47^-s + ⋯

L(s) = 1

− 0.0925·5^-s − 0.377i·7^-s − 1.93i·11^-s + 0.977i·13^-s + 0.947i·17^-s + 1.25·19^-s − 0.183·23^-s − 0.991·25^-s + 0.595·29^-s − 0.0113i·31^-s + 0.0349i·35^-s + 1.63i·37^-s − 0.928i·41^-s + 0.173·43^-s + 0.993·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.717 + 0.696i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.717 + 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$6048$ = $2^{5} \cdot 3^{3} \cdot 7$
Sign:	$0.717 + 0.696i$
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:	$(2,\ 6048,\ (\ :1/2),\ 0.717 + 0.696i)$

Particular Values

$L(1)$	$\approx$	$1.867038700$
$L(\frac12)$	$\approx$	$1.867038700$
$L(\frac{3}{2})$		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 + 0.206T + 5T^{2} $
	11	$ 1 + 6.42iT - 11T^{2} $
	13	$ 1 - 3.52iT - 13T^{2} $
	17	$ 1 - 3.90iT - 17T^{2} $
	19	$ 1 - 5.48T + 19T^{2} $
	23	$ 1 + 0.880T + 23T^{2} $
	29	$ 1 - 3.20T + 29T^{2} $
	31	$ 1 + 0.0631iT - 31T^{2} $
	37	$ 1 - 9.91iT - 37T^{2} $
	41	$ 1 + 5.94iT - 41T^{2} $
	43	$ 1 - 1.13T + 43T^{2} $
	47	$ 1 - 6.81T + 47T^{2} $
	53	$ 1 - 12.6T + 53T^{2} $
	59	$ 1 - 4.23iT - 59T^{2} $
	61	$ 1 + 12.3iT - 61T^{2} $
	67	$ 1 - 5.74T + 67T^{2} $
	71	$ 1 + 10.1T + 71T^{2} $
	73	$ 1 - 8.27T + 73T^{2} $
	79	$ 1 - 4.86iT - 79T^{2} $
	83	$ 1 + 7.51iT - 83T^{2} $
	89	$ 1 - 4.44iT - 89T^{2} $
	97	$ 1 + 17.3T + 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

−8.105920723857975935377367541280, −7.29526112723319477894503388347, −6.50169088202607671336204253501, −5.89212413476262577771314490426, −5.24109856312745433450202902382, −4.13333062206529207916964027300, −3.62822750941996135635566599471, −2.80105454197074600415833283861, −1.59454616068606894597631461948, −0.62523370935428357487292441887, 0.864639602710108507520842806846, 2.11228398966069030606924494599, 2.74545491073473301698770710871, 3.79125353312346454530051624674, 4.59555649008572400172179479518, 5.31776663544548508178698921211, 5.86414572906030333306967123158, 7.02514361040933503572285743783, 7.39628749463718607086794604657, 7.970094388010561142410619158362

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.867038700\)
\(L(\frac12)\)	\(\approx\)	\(1.867038700\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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