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Properties

Label 1-143-143.42-r0-0-0
Degree $1$
Conductor $143$
Sign $0.999 + 0.000188i$
Analytic cond. $0.664089$
Root an. cond. $0.664089$
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  + (0.913 − 0.406i)2-s + (0.669 + 0.743i)3-s + (0.669 − 0.743i)4-s + (−0.809 + 0.587i)5-s + (0.913 + 0.406i)6-s + (0.669 − 0.743i)7-s + (0.309 − 0.951i)8-s + (−0.104 + 0.994i)9-s + (−0.5 + 0.866i)10-s + 12-s + (0.309 − 0.951i)14-s + (−0.978 − 0.207i)15-s + (−0.104 − 0.994i)16-s + (0.913 + 0.406i)17-s + (0.309 + 0.951i)18-s + (−0.978 + 0.207i)19-s + ⋯
L(s)  = 1  + (0.913 − 0.406i)2-s + (0.669 + 0.743i)3-s + (0.669 − 0.743i)4-s + (−0.809 + 0.587i)5-s + (0.913 + 0.406i)6-s + (0.669 − 0.743i)7-s + (0.309 − 0.951i)8-s + (−0.104 + 0.994i)9-s + (−0.5 + 0.866i)10-s + 12-s + (0.309 − 0.951i)14-s + (−0.978 − 0.207i)15-s + (−0.104 − 0.994i)16-s + (0.913 + 0.406i)17-s + (0.309 + 0.951i)18-s + (−0.978 + 0.207i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.000188i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.000188i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.999 + 0.000188i$
Analytic conductor: \(0.664089\)
Root analytic conductor: \(0.664089\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (42, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 143,\ (0:\ ),\ 0.999 + 0.000188i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.049714791 + 0.0001931019070i\)
\(L(\frac12)\) \(\approx\) \(2.049714791 + 0.0001931019070i\)
\(L(1)\) \(\approx\) \(1.864221606 - 0.03268103251i\)
\(L(1)\) \(\approx\) \(1.864221606 - 0.03268103251i\)

Euler product

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

−28.36900855664370734620474447297, −27.22916351909909406378227751700, −25.925632498296931202824549400974, −25.08209924510384824035649476469, −24.21282640426210924240933606303, −23.72339947311029804851041124288, −22.59023732491488863975913464423, −21.15851563337533035977687856763, −20.547265640182470366365847020637, −19.44880610663378921720703929255, −18.35553223558955709456893192376, −17.03610223141460380663827129174, −15.83829518175030424230659432631, −14.87373104114848215734499078567, −14.1716247354790474595794918645, −12.73654115218462514903052209456, −12.288038198756491735940325967351, −11.21552989820475316798094625731, −8.89704953252529049225696514576, −8.11029217233470194082204260294, −7.2102780642977094684331959573, −5.78856557961830461061743077532, −4.52304744940624065172158291402, −3.241768781099968443103526007792, −1.866112208103397895401479211, 2.01373714220077335367070326843, 3.58734302442645209608662928856, 4.055942106798863021606040758419, 5.4242558675520829492662520060, 7.16883558825074512302467811363, 8.15183663890480901823303853870, 9.939644378778813773283027348472, 10.78568864693392321319572040740, 11.61074140914501886616888793103, 13.083545719142929488229986965459, 14.30670510552270939960162626592, 14.771743781478938315454162413651, 15.74083851382916959879909773144, 16.87254483930570450862497816913, 18.77373218997839024399015641190, 19.61348765960596026242242350472, 20.46950095925797515900605070653, 21.29861956664167934322839431612, 22.26021167749742968851856058806, 23.30562413584905107360896876060, 24.01187345639553101597384804992, 25.39725880342944619668475243866, 26.35625082263200551823470828646, 27.508473119731723583152559918114, 27.99416127342209017329455477429

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