LMFDB - L-function 2-1152-24.11-c1-0-3

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

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

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

Invariants

Degree:	$2$
Conductor:	$1152$ = $2^{7} \cdot 3^{2}$
Sign:	$-0.169 - 0.985i$
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:	$(2,\ 1152,\ (\ :1/2),\ -0.169 - 0.985i)$

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

−9.937398114102437259273349948833, −9.101130733120499954543010150139, −8.719045569866943114778596475735, −7.40855596434134233529865456160, −6.70982732685936686889925617239, −5.73557438393853998004627628029, −5.01846321955504386714905129998, −3.96013668216392425451004654620, −2.48149297543827828716775791813, −1.82779746240293080519351279816, 0.61226285962704655126300726828, 2.06509222166753356777367732558, 3.40199132105841922762692964135, 4.21136281533773992429205884677, 5.47145482224507658072708235653, 6.13685179323515020618591546317, 7.05179056661844512749662949148, 7.983828528182756081605787443917, 8.714805736115032370296000914526, 9.657273336235185664958547893612

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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