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Properties

Label 1-30e2-900.203-r1-0-0
Degree $1$
Conductor $900$
Sign $0.00698 - 0.999i$
Analytic cond. $96.7184$
Root an. cond. $96.7184$
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  + (−0.866 − 0.5i)7-s + (0.913 − 0.406i)11-s + (−0.406 + 0.913i)13-s + (0.951 + 0.309i)17-s + (0.309 − 0.951i)19-s + (−0.994 − 0.104i)23-s + (−0.978 − 0.207i)29-s + (0.978 − 0.207i)31-s + (0.587 + 0.809i)37-s + (−0.913 − 0.406i)41-s + (0.866 + 0.5i)43-s + (−0.207 + 0.978i)47-s + (0.5 + 0.866i)49-s + (0.951 − 0.309i)53-s + (−0.913 − 0.406i)59-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)7-s + (0.913 − 0.406i)11-s + (−0.406 + 0.913i)13-s + (0.951 + 0.309i)17-s + (0.309 − 0.951i)19-s + (−0.994 − 0.104i)23-s + (−0.978 − 0.207i)29-s + (0.978 − 0.207i)31-s + (0.587 + 0.809i)37-s + (−0.913 − 0.406i)41-s + (0.866 + 0.5i)43-s + (−0.207 + 0.978i)47-s + (0.5 + 0.866i)49-s + (0.951 − 0.309i)53-s + (−0.913 − 0.406i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00698 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00698 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.00698 - 0.999i$
Analytic conductor: \(96.7184\)
Root analytic conductor: \(96.7184\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (203, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 900,\ (1:\ ),\ 0.00698 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.018836277 - 1.011748171i\)
\(L(\frac12)\) \(\approx\) \(1.018836277 - 1.011748171i\)
\(L(1)\) \(\approx\) \(0.9726172321 - 0.1364708333i\)
\(L(1)\) \(\approx\) \(0.9726172321 - 0.1364708333i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (0.913 - 0.406i)T \)
13 \( 1 + (-0.406 + 0.913i)T \)
17 \( 1 + (0.951 + 0.309i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (-0.994 - 0.104i)T \)
29 \( 1 + (-0.978 - 0.207i)T \)
31 \( 1 + (0.978 - 0.207i)T \)
37 \( 1 + (0.587 + 0.809i)T \)
41 \( 1 + (-0.913 - 0.406i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 + (-0.207 + 0.978i)T \)
53 \( 1 + (0.951 - 0.309i)T \)
59 \( 1 + (-0.913 - 0.406i)T \)
61 \( 1 + (0.913 - 0.406i)T \)
67 \( 1 + (-0.207 - 0.978i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (0.587 - 0.809i)T \)
79 \( 1 + (-0.978 - 0.207i)T \)
83 \( 1 + (-0.743 + 0.669i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 + (-0.207 + 0.978i)T \)
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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

−22.13179592986318913860789196846, −21.18346541139503708742374872072, −20.19053871884882214681110078584, −19.68105230502107202525877595846, −18.74380055123073588548275740737, −18.103902719914238122255832016961, −17.05960211569368540312344285679, −16.42768397900741103851402770070, −15.55208257038647240258227438453, −14.76123117868846458698055491955, −14.007926316569566279421037881856, −12.93861284368206664940151816850, −12.21109926891044154002248656062, −11.69961756391122487113058843361, −10.21191387954984696037233201433, −9.850855189892340718878635191464, −8.90670265168580846571502208220, −7.87680801372573706266557520809, −7.0508276089777936040968433635, −5.9778542228789760990884001308, −5.41404263888807176732429873978, −4.02864141976161907272838297423, −3.26745078613217380946464383456, −2.20623248036678209586539284563, −0.94607428408333376719184801363, 0.36985530653372818003820571241, 1.490630930774169574089797799543, 2.79989997101829423224367562831, 3.75358601735588874238718459184, 4.512254069222001631641250673901, 5.8430615008127025878871946409, 6.568695664235972465944942581200, 7.355136281132927834229955233614, 8.405674903752645687333274178055, 9.51664528117599250480925102767, 9.843043539773249209772767447009, 11.07952392311627650292942027954, 11.8356081131642527429999083409, 12.63370671422946878216428399804, 13.67613998396869226089673498959, 14.150060767393141441293199863200, 15.160070368809496686064059944519, 16.135434764790909797848096032966, 16.78725012536006222223334207413, 17.341937085440893527466122202486, 18.5880921648039319102259989993, 19.25754407521807293608729116142, 19.82204077566383615383362541961, 20.70551938257408212481208239679, 21.73320097813283684140245105835

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