LMFDB - L-function 2-1152-24.11-c1-0-1

Dirichlet series

L(s) = 1

− 1.41·5^-s + 6i·13^-s − 4.24i·17^-s − 2.99·25^-s − 9.89·29^-s + 12i·37^-s + 12.7i·41^-s + 7·49^-s − 7.07·53^-s + 12i·61^-s − 8.48i·65^-s − 16·73^-s + 6i·85^-s + 4.24i·89^-s + 8·97^-s + ⋯

L(s) = 1

− 0.632·5^-s + 1.66i·13^-s − 1.02i·17^-s − 0.599·25^-s − 1.83·29^-s + 1.97i·37^-s + 1.98i·41^-s + 49^-s − 0.971·53^-s + 1.53i·61^-s − 1.05i·65^-s − 1.87·73^-s + 0.650i·85^-s + 0.449i·89^-s + 0.812·97^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1152$ = $2^{7} \cdot 3^{2}$
Sign:	$-0.577 - 0.816i$
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:	$(2,\ 1152,\ (\ :1/2),\ -0.577 - 0.816i)$

Particular Values

$L(1)$	$\approx$	$0.7141609417$
$L(\frac12)$	$\approx$	$0.7141609417$
$L(\frac{3}{2})$		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 - 7T^{2} $
	11	$ 1 - 11T^{2} $
	13	$ 1 - 6iT - 13T^{2} $
	17	$ 1 + 4.24iT - 17T^{2} $
	19	$ 1 + 19T^{2} $
	23	$ 1 + 23T^{2} $
	29	$ 1 + 9.89T + 29T^{2} $
	31	$ 1 - 31T^{2} $
	37	$ 1 - 12iT - 37T^{2} $
	41	$ 1 - 12.7iT - 41T^{2} $
	43	$ 1 + 43T^{2} $
	47	$ 1 + 47T^{2} $
	53	$ 1 + 7.07T + 53T^{2} $
	59	$ 1 - 59T^{2} $
	61	$ 1 - 12iT - 61T^{2} $
	67	$ 1 + 67T^{2} $
	71	$ 1 + 71T^{2} $
	73	$ 1 + 16T + 73T^{2} $
	79	$ 1 - 79T^{2} $
	83	$ 1 - 83T^{2} $
	89	$ 1 - 4.24iT - 89T^{2} $
	97	$ 1 - 8T + 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

−9.870246114662806891845548087738, −9.339143069543543640404899612567, −8.457533985937668771308890300819, −7.52996166570551465285248906583, −6.89745690342927009606610399059, −5.93574652185219288794978444693, −4.74533844902306284267694050419, −4.07530358119123125094366381344, −2.93349183556305082939007592302, −1.59120727489496649931361577032, 0.30595094782506241078071687660, 2.03394373135791130806666996269, 3.42612492911108682584712036061, 4.04312779743097331899890777333, 5.42256329844691614814054598713, 5.92803941546673238328386711729, 7.31752185705540260972706185831, 7.76654095292086480595636461317, 8.616893342732322462472511079730, 9.478899668879200354396405190503

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.7141609417\)
\(L(\frac12)\)	\(\approx\)	\(0.7141609417\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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