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Properties

Label 1-483-483.44-r0-0-0
Degree $1$
Conductor $483$
Sign $0.815 + 0.578i$
Analytic cond. $2.24304$
Root an. cond. $2.24304$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.580 + 0.814i)2-s + (−0.327 − 0.945i)4-s + (0.0475 − 0.998i)5-s + (0.959 + 0.281i)8-s + (0.786 + 0.618i)10-s + (0.580 + 0.814i)11-s + (−0.142 + 0.989i)13-s + (−0.786 + 0.618i)16-s + (0.981 − 0.189i)17-s + (−0.981 − 0.189i)19-s + (−0.959 + 0.281i)20-s − 22-s + (−0.995 − 0.0950i)25-s + (−0.723 − 0.690i)26-s + (0.654 + 0.755i)29-s + ⋯
L(s)  = 1  + (−0.580 + 0.814i)2-s + (−0.327 − 0.945i)4-s + (0.0475 − 0.998i)5-s + (0.959 + 0.281i)8-s + (0.786 + 0.618i)10-s + (0.580 + 0.814i)11-s + (−0.142 + 0.989i)13-s + (−0.786 + 0.618i)16-s + (0.981 − 0.189i)17-s + (−0.981 − 0.189i)19-s + (−0.959 + 0.281i)20-s − 22-s + (−0.995 − 0.0950i)25-s + (−0.723 − 0.690i)26-s + (0.654 + 0.755i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.815 + 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.815 + 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.815 + 0.578i$
Analytic conductor: \(2.24304\)
Root analytic conductor: \(2.24304\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 483,\ (0:\ ),\ 0.815 + 0.578i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9538172648 + 0.3037628459i\)
\(L(\frac12)\) \(\approx\) \(0.9538172648 + 0.3037628459i\)
\(L(1)\) \(\approx\) \(0.8217430288 + 0.1944051322i\)
\(L(1)\) \(\approx\) \(0.8217430288 + 0.1944051322i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.580 + 0.814i)T \)
5 \( 1 + (0.0475 - 0.998i)T \)
11 \( 1 + (0.580 + 0.814i)T \)
13 \( 1 + (-0.142 + 0.989i)T \)
17 \( 1 + (0.981 - 0.189i)T \)
19 \( 1 + (-0.981 - 0.189i)T \)
29 \( 1 + (0.654 + 0.755i)T \)
31 \( 1 + (0.723 - 0.690i)T \)
37 \( 1 + (0.888 + 0.458i)T \)
41 \( 1 + (-0.841 - 0.540i)T \)
43 \( 1 + (0.959 - 0.281i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (0.928 + 0.371i)T \)
59 \( 1 + (0.786 + 0.618i)T \)
61 \( 1 + (-0.235 - 0.971i)T \)
67 \( 1 + (0.995 + 0.0950i)T \)
71 \( 1 + (-0.415 - 0.909i)T \)
73 \( 1 + (-0.327 - 0.945i)T \)
79 \( 1 + (-0.928 + 0.371i)T \)
83 \( 1 + (0.841 - 0.540i)T \)
89 \( 1 + (0.723 + 0.690i)T \)
97 \( 1 + (-0.841 - 0.540i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.30160581836964481325755355334, −22.742527774248997731164572402751, −21.70014822124312930230046826639, −21.33386618089227456201117509194, −20.124421846655562597207291920183, −19.25674921706689507030059357788, −18.81405676408610202659653756828, −17.801252518289522136009610339111, −17.145462305179961006240746225013, −16.14565157802520734182055625749, −14.96029808519916589447253852648, −14.094808576521271010478883446993, −13.14829036896692938728591042033, −12.118519430192020395842405673540, −11.31761863362484886712696734178, −10.41431537821988031484987344140, −9.91340367173995575781512362563, −8.58523843923603132744149907963, −7.885530428359926447384592296744, −6.77361053953905213136303235688, −5.706698207937698405744716696630, −4.101006663769077931950607317656, −3.22029844205420562256625636484, −2.390651997419119369387490759723, −0.919629897879572372044045090585, 1.01174201557543611728828673771, 2.07114498223203549663933842819, 4.1778314064753532328607793041, 4.81939790462766926984048015737, 5.9554987748047131723582628744, 6.88904699727641999098607450063, 7.86372897704021161488120660425, 8.83553948500553874276181856271, 9.45897843188049012823739408391, 10.33050159419030706662524253811, 11.691674896644352102457061240295, 12.54326478800450874957603864358, 13.667693089024235766660226048279, 14.49426348571074511307043136330, 15.3704826240790932525262136333, 16.355085139272586845057931480364, 16.957626388039653413876287441479, 17.56711228704765050423934180329, 18.74022478454781001996952587368, 19.45643954699141528246846160790, 20.30956338113354405605890803881, 21.20275342365597834875755954928, 22.323197149782355990444149104748, 23.45093123387656738399582676201, 23.83239973688083033205754141095

Graph of the $Z$-function along the critical line