LMFDB - L-function 2-6048-12.11-c1-0-72

Dirichlet series

L(s) = 1

− 1.93i·5^-s + i·7^-s + 1.13·11^-s − 3.19·13^-s + 5.93i·17^-s + 1.33i·19^-s − 2.96·23^-s + 1.26·25^-s + 1.65i·29^-s − 10.3i·31^-s + 1.93·35^-s + 0.538·37^-s − 9.51i·41^-s + 6.65i·43^-s − 11.1·47^-s + ⋯

L(s) = 1

− 0.863i·5^-s + 0.377i·7^-s + 0.342·11^-s − 0.885·13^-s + 1.43i·17^-s + 0.307i·19^-s − 0.618·23^-s + 0.253·25^-s + 0.307i·29^-s − 1.86i·31^-s + 0.326·35^-s + 0.0884·37^-s − 1.48i·41^-s + 1.01i·43^-s − 1.62·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$6048$ = $2^{5} \cdot 3^{3} \cdot 7$
Sign:	$-0.707 + 0.707i$
Analytic conductor:	$48.2935$
Root analytic conductor:	$6.94935$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{6048} (2591, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 6048,\ (\ :1/2),\ -0.707 + 0.707i)$

Particular Values

$L(1)$	$\approx$	$0.8477029638$
$L(\frac12)$	$\approx$	$0.8477029638$
$L(\frac{3}{2})$		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 + 1.93iT - 5T^{2} $
	11	$ 1 - 1.13T + 11T^{2} $
	13	$ 1 + 3.19T + 13T^{2} $
	17	$ 1 - 5.93iT - 17T^{2} $
	19	$ 1 - 1.33iT - 19T^{2} $
	23	$ 1 + 2.96T + 23T^{2} $
	29	$ 1 - 1.65iT - 29T^{2} $
	31	$ 1 + 10.3iT - 31T^{2} $
	37	$ 1 - 0.538T + 37T^{2} $
	41	$ 1 + 9.51iT - 41T^{2} $
	43	$ 1 - 6.65iT - 43T^{2} $
	47	$ 1 + 11.1T + 47T^{2} $
	53	$ 1 + 1.68iT - 53T^{2} $
	59	$ 1 - 1.82T + 59T^{2} $
	61	$ 1 - 14.1T + 61T^{2} $
	67	$ 1 + 9.53iT - 67T^{2} $
	71	$ 1 + 10.9T + 71T^{2} $
	73	$ 1 - 0.319T + 73T^{2} $
	79	$ 1 + 12.9iT - 79T^{2} $
	83	$ 1 + 3.24T + 83T^{2} $
	89	$ 1 + 13.7iT - 89T^{2} $
	97	$ 1 - 2.33T + 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

−7.972237962228745067736578536818, −7.16909106900430738768937428510, −6.21316821316653936844160228914, −5.73564054099630975269334886980, −4.86799628869235625631142932532, −4.23830060680147877428734788353, −3.43497846045044207092216055514, −2.25691247649721364447241371658, −1.54441485892614483064867657993, −0.22187799429114881856570736105, 1.14196646124642502667827024022, 2.43646209553396556481426927927, 2.99469685461390323352681921334, 3.87484983298682769698770803011, 4.82610505832826569727319168592, 5.33482909048169141434671519192, 6.56980135915698850939720775641, 6.84301141552312589081121494788, 7.46861934237490080612335208333, 8.256502274742312880780822581659

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.8477029638\)
\(L(\frac12)\)	\(\approx\)	\(0.8477029638\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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