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Properties

Label 1-483-483.383-r1-0-0
Degree $1$
Conductor $483$
Sign $0.784 + 0.619i$
Analytic cond. $51.9055$
Root an. cond. $51.9055$
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  + (−0.235 − 0.971i)2-s + (−0.888 + 0.458i)4-s + (−0.928 + 0.371i)5-s + (0.654 + 0.755i)8-s + (0.580 + 0.814i)10-s + (0.235 − 0.971i)11-s + (−0.415 + 0.909i)13-s + (0.580 − 0.814i)16-s + (−0.0475 − 0.998i)17-s + (0.0475 − 0.998i)19-s + (0.654 − 0.755i)20-s − 22-s + (0.723 − 0.690i)25-s + (0.981 + 0.189i)26-s + (−0.841 + 0.540i)29-s + ⋯
L(s)  = 1  + (−0.235 − 0.971i)2-s + (−0.888 + 0.458i)4-s + (−0.928 + 0.371i)5-s + (0.654 + 0.755i)8-s + (0.580 + 0.814i)10-s + (0.235 − 0.971i)11-s + (−0.415 + 0.909i)13-s + (0.580 − 0.814i)16-s + (−0.0475 − 0.998i)17-s + (0.0475 − 0.998i)19-s + (0.654 − 0.755i)20-s − 22-s + (0.723 − 0.690i)25-s + (0.981 + 0.189i)26-s + (−0.841 + 0.540i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.784 + 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.784 + 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.784 + 0.619i$
Analytic conductor: \(51.9055\)
Root analytic conductor: \(51.9055\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 483,\ (1:\ ),\ 0.784 + 0.619i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5266626674 + 0.1829710868i\)
\(L(\frac12)\) \(\approx\) \(0.5266626674 + 0.1829710868i\)
\(L(1)\) \(\approx\) \(0.6118284725 - 0.2380202189i\)
\(L(1)\) \(\approx\) \(0.6118284725 - 0.2380202189i\)

Euler product

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

−23.6123275092642088927216211196, −22.75582137621349711059923323855, −22.24628945910124899796684562227, −20.74648447235291375616399404168, −19.9066188184032949230800881278, −19.15401359545819731668595365391, −18.23199965801203001396597892828, −17.255397112345959369529184825750, −16.67377196040481012865990452723, −15.599485544841568075897232195146, −15.05043943221700656687243921024, −14.33533822063910680516775822794, −12.820490854382284363675689891525, −12.53480388405013078027195984520, −11.09103210198206034856965862203, −10.014688939649163206254781332040, −9.170625460851831695100907780701, −7.907698819189818299653764201455, −7.72582541847157684774333373580, −6.45295361399803168119487412704, −5.40103006710681338553087653189, −4.42218923350930508883201076546, −3.58064405689497745623839800433, −1.62646919401081708945040714443, −0.22072148735592259573967398708, 0.829475363790983710137298944757, 2.3604034570851814697073197549, 3.31913678429134552828414438332, 4.1953073372760794485271411071, 5.22296897252082111929473164306, 6.84798122877792552301254819812, 7.69522490039201006933995976639, 8.83080586431498816525493539645, 9.43499184992007514836250310605, 10.8085738309761618324394778879, 11.35761157583025205514058557785, 11.99018074068477093057095226394, 13.101631345449350784001830874752, 14.02429671690845071811236952455, 14.82602108313471780718376724480, 16.170809442581239676169343739804, 16.740177372179079370524888599487, 18.068080303015830057153343119, 18.65848164372925298467266790483, 19.525679761439457690100271215, 19.99079553446585403732651017880, 21.105422861218439001278084504193, 21.96845601507967350438837234851, 22.50587075748131553798877014043, 23.6115710070119240578563383219

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