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Properties

Label 1-143-143.29-r1-0-0
Degree $1$
Conductor $143$
Sign $0.735 - 0.677i$
Analytic cond. $15.3674$
Root an. cond. $15.3674$
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  + (0.978 + 0.207i)2-s + (0.913 − 0.406i)3-s + (0.913 + 0.406i)4-s + (0.309 − 0.951i)5-s + (0.978 − 0.207i)6-s + (−0.913 − 0.406i)7-s + (0.809 + 0.587i)8-s + (0.669 − 0.743i)9-s + (0.5 − 0.866i)10-s + 12-s + (−0.809 − 0.587i)14-s + (−0.104 − 0.994i)15-s + (0.669 + 0.743i)16-s + (0.978 − 0.207i)17-s + (0.809 − 0.587i)18-s + (0.104 − 0.994i)19-s + ⋯
L(s)  = 1  + (0.978 + 0.207i)2-s + (0.913 − 0.406i)3-s + (0.913 + 0.406i)4-s + (0.309 − 0.951i)5-s + (0.978 − 0.207i)6-s + (−0.913 − 0.406i)7-s + (0.809 + 0.587i)8-s + (0.669 − 0.743i)9-s + (0.5 − 0.866i)10-s + 12-s + (−0.809 − 0.587i)14-s + (−0.104 − 0.994i)15-s + (0.669 + 0.743i)16-s + (0.978 − 0.207i)17-s + (0.809 − 0.587i)18-s + (0.104 − 0.994i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.735 - 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.735 - 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.735 - 0.677i$
Analytic conductor: \(15.3674\)
Root analytic conductor: \(15.3674\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 143,\ (1:\ ),\ 0.735 - 0.677i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.200477779 - 1.638399186i\)
\(L(\frac12)\) \(\approx\) \(4.200477779 - 1.638399186i\)
\(L(1)\) \(\approx\) \(2.517075101 - 0.5071180270i\)
\(L(1)\) \(\approx\) \(2.517075101 - 0.5071180270i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.978 + 0.207i)T \)
3 \( 1 + (0.913 - 0.406i)T \)
5 \( 1 + (0.309 - 0.951i)T \)
7 \( 1 + (-0.913 - 0.406i)T \)
17 \( 1 + (0.978 - 0.207i)T \)
19 \( 1 + (0.104 - 0.994i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.104 + 0.994i)T \)
31 \( 1 + (0.309 + 0.951i)T \)
37 \( 1 + (-0.104 - 0.994i)T \)
41 \( 1 + (-0.913 + 0.406i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (-0.809 - 0.587i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (0.913 + 0.406i)T \)
61 \( 1 + (0.978 - 0.207i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.978 + 0.207i)T \)
73 \( 1 + (0.809 - 0.587i)T \)
79 \( 1 + (-0.309 - 0.951i)T \)
83 \( 1 + (-0.309 + 0.951i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (0.669 - 0.743i)T \)
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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

−28.37248314830797173373580972096, −27.02108720659230123344036271270, −25.770382704561328076372772448674, −25.465275964661057147510924879877, −24.287344962569638790190093444203, −22.78717906635545687008796634416, −22.29007296848907175037613921538, −21.26993246799346011083356181751, −20.46875667506887958390335384639, −19.14224431623371461425699986200, −18.7680912363026199828554480610, −16.63841239760023206966955918338, −15.59936101138580924600898327513, −14.766104138774527596190641799018, −13.98492935218703199466467414941, −13.018330269315751741036227322427, −11.823589666163398143318567349893, −10.25696572479450289698857383188, −9.85308612912031550725738647349, −8.00809062808935874381575007387, −6.69719759958708498165714562023, −5.65429713383079518562484773246, −3.942443669285619909468182310188, −3.08392073973293100829331083088, −2.096077616471690955366696051627, 1.2941244673100983730508148317, 2.85245337111281502467113002038, 3.91203388184282661122654671926, 5.29659347424948465013738078190, 6.64379271791021899115405530389, 7.6604391974053215836946636076, 8.940843470253624121593594363629, 10.09667214387707605623594610150, 11.95362695679929239443847630440, 12.8629133983784491280839791359, 13.50436967090618120326260129794, 14.39375044163493263603834626716, 15.73246649764482017720444598137, 16.435526757506027221973993686268, 17.74199715737795305812194357930, 19.48330336882719784510374289964, 19.990985531784482844875390092010, 20.99725456403112042628365611477, 21.84215501256502652861770393889, 23.30599190499828571902864885669, 23.93459888413805591505298938826, 25.04939824411002486153661786527, 25.56705683538424164289454396454, 26.51679397837374638357482420721, 28.17349925934869804098145498107

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