LMFDB - L-function 2-1152-24.11-c1-0-8

Dirichlet series

L(s) = 1

− 1.41·5^-s − 2.82i·7^-s + 4i·11^-s + 2i·13^-s + 1.41i·17^-s + 5.65·19^-s + 4·23^-s − 2.99·25^-s + 7.07·29^-s − 8.48i·31^-s + 4.00i·35^-s − 8i·37^-s − 4.24i·41^-s + 11.3·43^-s + 12·47^-s + ⋯

L(s) = 1

− 0.632·5^-s − 1.06i·7^-s + 1.20i·11^-s + 0.554i·13^-s + 0.342i·17^-s + 1.29·19^-s + 0.834·23^-s − 0.599·25^-s + 1.31·29^-s − 1.52i·31^-s + 0.676i·35^-s − 1.31i·37^-s − 0.662i·41^-s + 1.72·43^-s + 1.75·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1152$ = $2^{7} \cdot 3^{2}$
Sign:	$0.985 + 0.169i$
Analytic conductor:	$9.19876$
Root analytic conductor:	$3.03294$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1152} (575, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1152,\ (\ :1/2),\ 0.985 + 0.169i)$

Particular Values

$L(1)$	$\approx$	$1.463246573$
$L(\frac12)$	$\approx$	$1.463246573$
$L(\frac{3}{2})$		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 $
good	5	$ 1 + 1.41T + 5T^{2} $
	7	$ 1 + 2.82iT - 7T^{2} $
	11	$ 1 - 4iT - 11T^{2} $
	13	$ 1 - 2iT - 13T^{2} $
	17	$ 1 - 1.41iT - 17T^{2} $
	19	$ 1 - 5.65T + 19T^{2} $
	23	$ 1 - 4T + 23T^{2} $
	29	$ 1 - 7.07T + 29T^{2} $
	31	$ 1 + 8.48iT - 31T^{2} $
	37	$ 1 + 8iT - 37T^{2} $
	41	$ 1 + 4.24iT - 41T^{2} $
	43	$ 1 - 11.3T + 43T^{2} $
	47	$ 1 - 12T + 47T^{2} $
	53	$ 1 + 12.7T + 53T^{2} $
	59	$ 1 - 59T^{2} $
	61	$ 1 - 8iT - 61T^{2} $
	67	$ 1 - 5.65T + 67T^{2} $
	71	$ 1 - 4T + 71T^{2} $
	73	$ 1 - 8T + 73T^{2} $
	79	$ 1 + 2.82iT - 79T^{2} $
	83	$ 1 - 12iT - 83T^{2} $
	89	$ 1 - 15.5iT - 89T^{2} $
	97	$ 1 + 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

−9.721149170636961364874190384203, −9.131574402576869388418047435753, −7.77446415581598352434100362066, −7.46739265263169160223209048700, −6.66862840200827491969427911818, −5.43740440800095212987232032961, −4.29821067243884985886996079293, −3.89360831860130700146941109336, −2.42299520525990374527095817847, −0.919482224341953366354628498840, 0.963600920384473100458553095166, 2.80770860167350525144284991224, 3.34715330281123951424148908266, 4.78903020089471820399195516706, 5.54879725037018981711901761452, 6.38345434845708658101467688378, 7.48409695275411928779406154145, 8.265944888087932159690457967210, 8.881179420418852309059662427962, 9.708632250683248754805940447698

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.463246573\)
\(L(\frac12)\)	\(\approx\)	\(1.463246573\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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