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Properties

Label 1-143-143.3-r0-0-0
Degree $1$
Conductor $143$
Sign $0.0696 + 0.997i$
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  + (0.669 + 0.743i)2-s + (−0.104 − 0.994i)3-s + (−0.104 + 0.994i)4-s + (0.309 + 0.951i)5-s + (0.669 − 0.743i)6-s + (−0.104 + 0.994i)7-s + (−0.809 + 0.587i)8-s + (−0.978 + 0.207i)9-s + (−0.5 + 0.866i)10-s + 12-s + (−0.809 + 0.587i)14-s + (0.913 − 0.406i)15-s + (−0.978 − 0.207i)16-s + (0.669 − 0.743i)17-s + (−0.809 − 0.587i)18-s + (0.913 + 0.406i)19-s + ⋯
L(s)  = 1  + (0.669 + 0.743i)2-s + (−0.104 − 0.994i)3-s + (−0.104 + 0.994i)4-s + (0.309 + 0.951i)5-s + (0.669 − 0.743i)6-s + (−0.104 + 0.994i)7-s + (−0.809 + 0.587i)8-s + (−0.978 + 0.207i)9-s + (−0.5 + 0.866i)10-s + 12-s + (−0.809 + 0.587i)14-s + (0.913 − 0.406i)15-s + (−0.978 − 0.207i)16-s + (0.669 − 0.743i)17-s + (−0.809 − 0.587i)18-s + (0.913 + 0.406i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0696 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0696 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.0696 + 0.997i$
Analytic conductor: \(0.664089\)
Root analytic conductor: \(0.664089\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 143,\ (0:\ ),\ 0.0696 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.035438210 + 0.9656500391i\)
\(L(\frac12)\) \(\approx\) \(1.035438210 + 0.9656500391i\)
\(L(1)\) \(\approx\) \(1.209495640 + 0.6058031547i\)
\(L(1)\) \(\approx\) \(1.209495640 + 0.6058031547i\)

Euler product

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

−28.32038380964970466166635655132, −27.311822912461693727323750228010, −26.350819031900377855282767947872, −24.96388493211570978712980066896, −23.74832963529411521210018583199, −23.05073345450511961602110551308, −21.84866795645795947249308849630, −21.16550103624072676205015073427, −20.17673033407253321586437779925, −19.77185576517139162416697098864, −17.93738908587138614738575516908, −16.726460748374393563274312949334, −15.945764291638957715260302000935, −14.581937768276154081400778329880, −13.73615475228767550528884758470, −12.62309839376540034584558398059, −11.50937214607958666264126634830, −10.29537369826723423728813788792, −9.72855290502328867688748167577, −8.39820815676684605582812375205, −6.290935908294679047237156417031, −5.043881461501422583628053255803, −4.29402796528333613233652538521, −3.10452718485932858430581726063, −1.12959939951812032535919366595, 2.29506060886130325471774366971, 3.29039233855778535780545019036, 5.41653914289773927622637234859, 6.1115174992545477559627099743, 7.206661231059254017005541025653, 8.09691190138632694520920772131, 9.56732553103984307761027835405, 11.528625175267360170949126986669, 12.12139053172585571148548960309, 13.43488435254764592741230772228, 14.16512072136807944256061743082, 15.139654955258483588182052175572, 16.29019139315212508169755718795, 17.677893362806011369867892608469, 18.25603931057825253520009264512, 19.21338095022689389803425088678, 20.84032613010938310811849303033, 22.04098156402877798349001122561, 22.665840828657845144768519257168, 23.59097484534268352639683112449, 24.74166198578811618463964544640, 25.31646967717628773777867828943, 26.10291375929875243335792197269, 27.37289402413832324554778409765, 28.8917912959496902113905937341

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