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Properties

Label 1-483-483.416-r0-0-0
Degree $1$
Conductor $483$
Sign $0.800 + 0.599i$
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  + (0.995 + 0.0950i)2-s + (0.981 + 0.189i)4-s + (−0.888 + 0.458i)5-s + (0.959 + 0.281i)8-s + (−0.928 + 0.371i)10-s + (0.995 − 0.0950i)11-s + (0.142 − 0.989i)13-s + (0.928 + 0.371i)16-s + (−0.327 + 0.945i)17-s + (0.327 + 0.945i)19-s + (−0.959 + 0.281i)20-s + 22-s + (0.580 − 0.814i)25-s + (0.235 − 0.971i)26-s + (0.654 + 0.755i)29-s + ⋯
L(s)  = 1  + (0.995 + 0.0950i)2-s + (0.981 + 0.189i)4-s + (−0.888 + 0.458i)5-s + (0.959 + 0.281i)8-s + (−0.928 + 0.371i)10-s + (0.995 − 0.0950i)11-s + (0.142 − 0.989i)13-s + (0.928 + 0.371i)16-s + (−0.327 + 0.945i)17-s + (0.327 + 0.945i)19-s + (−0.959 + 0.281i)20-s + 22-s + (0.580 − 0.814i)25-s + (0.235 − 0.971i)26-s + (0.654 + 0.755i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.800 + 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.800 + 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.800 + 0.599i$
Analytic conductor: \(2.24304\)
Root analytic conductor: \(2.24304\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (416, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 483,\ (0:\ ),\ 0.800 + 0.599i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.265646357 + 0.7548786419i\)
\(L(\frac12)\) \(\approx\) \(2.265646357 + 0.7548786419i\)
\(L(1)\) \(\approx\) \(1.777681993 + 0.3207304570i\)
\(L(1)\) \(\approx\) \(1.777681993 + 0.3207304570i\)

Euler product

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

−23.475077829965875779366650162266, −23.000674960263587139461798009297, −22.01307714476861250611659568201, −21.27902244537514109597942751151, −20.230007480497115011459806977253, −19.72078285551608783659138957825, −18.94359284426082507575761552767, −17.5380313054996512638516534633, −16.40908159885710593898123429949, −15.94924043760478245917837944113, −14.99221328434861900272182663756, −14.10182061190929187761910812675, −13.34188609336744779860808791118, −12.17449374087007716227406261429, −11.7076896128179199878684673031, −10.96931880506605140429042354797, −9.51303522884129509079451100818, −8.59322951076353110825586738274, −7.20177534593502922107064375702, −6.73919015976584853956027389132, −5.328164766713915500705904800452, −4.415729046221730821679270955335, −3.760582348215809108077406474168, −2.50117457825736137038422464783, −1.1233791317634844029618623417, 1.451951761449186798834629209105, 2.97996417366131177194278411279, 3.70582723805192437381072175114, 4.57203861509716600530132820319, 5.886999869671349776969934883887, 6.623417494553412519032573354790, 7.69740641324333577116962842455, 8.42105760198667630722461561527, 10.08043766224237846693726626567, 10.96882815354492934035380173865, 11.76822467262430508324049772459, 12.515831138997689473105875129677, 13.440524502689622228955370249988, 14.675035681604152321805858949292, 14.8952569406995176540387409000, 15.97101932139422661087995105253, 16.704512659176970719782799648159, 17.81515304724936287939788959277, 19.03038634626467568356528283884, 19.81641218240295726447393819993, 20.397894728322724814170359177606, 21.55532637007542631277561225022, 22.40575786450439540661028174548, 22.845332965355546944930044049572, 23.7600679703855244360116500170

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