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Properties

Label 1-633-633.227-r1-0-0
Degree $1$
Conductor $633$
Sign $-0.682 - 0.731i$
Analytic cond. $68.0252$
Root an. cond. $68.0252$
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  + (−0.992 − 0.119i)2-s + (0.971 + 0.237i)4-s + (0.995 + 0.0896i)5-s + (−0.599 − 0.800i)7-s + (−0.936 − 0.351i)8-s + (−0.978 − 0.207i)10-s + (−0.858 − 0.512i)11-s + (−0.0448 − 0.998i)13-s + (0.5 + 0.866i)14-s + (0.887 + 0.460i)16-s + (−0.251 + 0.967i)17-s + (0.913 − 0.406i)19-s + (0.946 + 0.323i)20-s + (0.791 + 0.611i)22-s + (0.809 − 0.587i)23-s + ⋯
L(s)  = 1  + (−0.992 − 0.119i)2-s + (0.971 + 0.237i)4-s + (0.995 + 0.0896i)5-s + (−0.599 − 0.800i)7-s + (−0.936 − 0.351i)8-s + (−0.978 − 0.207i)10-s + (−0.858 − 0.512i)11-s + (−0.0448 − 0.998i)13-s + (0.5 + 0.866i)14-s + (0.887 + 0.460i)16-s + (−0.251 + 0.967i)17-s + (0.913 − 0.406i)19-s + (0.946 + 0.323i)20-s + (0.791 + 0.611i)22-s + (0.809 − 0.587i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 633 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.682 - 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 633 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.682 - 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(633\)    =    \(3 \cdot 211\)
Sign: $-0.682 - 0.731i$
Analytic conductor: \(68.0252\)
Root analytic conductor: \(68.0252\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{633} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 633,\ (1:\ ),\ -0.682 - 0.731i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3979554352 - 0.9156479946i\)
\(L(\frac12)\) \(\approx\) \(0.3979554352 - 0.9156479946i\)
\(L(1)\) \(\approx\) \(0.7124892970 - 0.2214878910i\)
\(L(1)\) \(\approx\) \(0.7124892970 - 0.2214878910i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
211 \( 1 \)
good2 \( 1 + (-0.992 - 0.119i)T \)
5 \( 1 + (0.995 + 0.0896i)T \)
7 \( 1 + (-0.599 - 0.800i)T \)
11 \( 1 + (-0.858 - 0.512i)T \)
13 \( 1 + (-0.0448 - 0.998i)T \)
17 \( 1 + (-0.251 + 0.967i)T \)
19 \( 1 + (0.913 - 0.406i)T \)
23 \( 1 + (0.809 - 0.587i)T \)
29 \( 1 + (0.842 - 0.538i)T \)
31 \( 1 + (0.365 + 0.930i)T \)
37 \( 1 + (0.0149 + 0.999i)T \)
41 \( 1 + (0.447 - 0.894i)T \)
43 \( 1 + (-0.988 - 0.149i)T \)
47 \( 1 + (0.646 - 0.762i)T \)
53 \( 1 + (-0.971 + 0.237i)T \)
59 \( 1 + (-0.998 + 0.0598i)T \)
61 \( 1 + (0.669 - 0.743i)T \)
67 \( 1 + (-0.222 - 0.974i)T \)
71 \( 1 + (-0.309 - 0.951i)T \)
73 \( 1 + (0.0747 + 0.997i)T \)
79 \( 1 + (0.936 - 0.351i)T \)
83 \( 1 + (-0.669 - 0.743i)T \)
89 \( 1 + (-0.753 + 0.657i)T \)
97 \( 1 + (0.134 - 0.990i)T \)
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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

−23.09194024056235605133939376173, −22.0285349268794556466898324783, −21.1855302614752826091005621991, −20.62843034002219600696505300743, −19.58970859685166281863515382366, −18.64067153918116799738054516244, −18.19098345564395458377060634504, −17.41261888342640868286766882207, −16.35653292399532604707369996045, −15.89547676037075895298568482959, −14.87924695147074577232586500198, −13.88201593366818415754294942489, −12.87224713465386130059328953643, −11.95088464137155414755828063184, −11.049194615305331259137830545203, −9.83436598148330411283002659127, −9.530036897255361973397790077752, −8.73234692559504129629266275703, −7.50218966473180200252050042428, −6.672655956976470970707772859995, −5.77636814920726736359673371641, −4.93904516225258988112239215511, −2.95307621931530076133541486100, −2.32504045704212815800211323963, −1.20545254710857370653718748168, 0.351048112841430382254049092482, 1.30084234206389701123058554667, 2.68524421387615481813822056037, 3.293503177136173038458207839456, 5.06775242425601905370637255272, 6.13307492678123229691482454890, 6.854176159523286804662091211894, 7.90183119518247117828879764704, 8.74984074379061361665554727510, 9.80169128770448791035687299637, 10.44653778248946565565555297195, 10.88130662408453935512698850982, 12.36343576815598399909993654375, 13.15495091050588680869269724183, 13.89562568705974104349144327068, 15.21867458460932182426446098035, 15.93582448584881037338169167629, 16.929510078147056324175375779, 17.43640377061138449250260380057, 18.24134484683720979918656146999, 19.04774252401517087986935904073, 19.96815887517435066722153621325, 20.61936110933345430716638864281, 21.435856027819640666990820219582, 22.247902329355883666292905521190

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