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Properties

Label 1-143-143.28-r0-0-0
Degree $1$
Conductor $143$
Sign $0.811 + 0.584i$
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  + (0.994 − 0.104i)2-s + (−0.978 − 0.207i)3-s + (0.978 − 0.207i)4-s + (−0.587 + 0.809i)5-s + (−0.994 − 0.104i)6-s + (0.207 + 0.978i)7-s + (0.951 − 0.309i)8-s + (0.913 + 0.406i)9-s + (−0.5 + 0.866i)10-s − 12-s + (0.309 + 0.951i)14-s + (0.743 − 0.669i)15-s + (0.913 − 0.406i)16-s + (−0.104 + 0.994i)17-s + (0.951 + 0.309i)18-s + (0.743 + 0.669i)19-s + ⋯
L(s)  = 1  + (0.994 − 0.104i)2-s + (−0.978 − 0.207i)3-s + (0.978 − 0.207i)4-s + (−0.587 + 0.809i)5-s + (−0.994 − 0.104i)6-s + (0.207 + 0.978i)7-s + (0.951 − 0.309i)8-s + (0.913 + 0.406i)9-s + (−0.5 + 0.866i)10-s − 12-s + (0.309 + 0.951i)14-s + (0.743 − 0.669i)15-s + (0.913 − 0.406i)16-s + (−0.104 + 0.994i)17-s + (0.951 + 0.309i)18-s + (0.743 + 0.669i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.811 + 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.811 + 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.811 + 0.584i$
Analytic conductor: \(0.664089\)
Root analytic conductor: \(0.664089\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 143,\ (0:\ ),\ 0.811 + 0.584i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.344566596 + 0.4340502796i\)
\(L(\frac12)\) \(\approx\) \(1.344566596 + 0.4340502796i\)
\(L(1)\) \(\approx\) \(1.337253646 + 0.1876604384i\)
\(L(1)\) \(\approx\) \(1.337253646 + 0.1876604384i\)

Euler product

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

−28.352258316514639020130230667875, −27.36133390670821088794093446639, −26.311795558145611222533964451563, −24.67930648190748304832047621830, −24.05120428857449458171438468066, −23.19493675073441857014519580203, −22.57734048800199417977871157393, −21.29697003478814796425638365859, −20.50572507671998484985942950611, −19.58189101653314284924991368443, −17.79794379266997643566623524117, −16.72100176430107917659772993385, −16.13553066371172210764561520124, −15.142851823729071326392369594572, −13.64481577442499908125425978573, −12.827688181925503821124640576048, −11.60982599965778211586693404850, −11.13803450031038918588906571183, −9.62976404407655701070307863822, −7.688231811023251274022698248307, −6.8559949472257607060924770703, −5.29965356063341506135347125720, −4.64267334592571957639540219606, −3.526805506646545197568023403814, −1.16827023557096334546852935345, 1.924866612465916313615014128897, 3.42328472019538165781056690161, 4.79663971839523299314653521762, 5.92296885560107091684658702411, 6.77329655926434908455057717903, 8.0236482844035081781488956744, 10.20356172400861185441557031993, 11.14164196852418722432075644255, 11.97491798696203555911425274559, 12.693746134836393526636269333003, 14.16793464616077246187963672501, 15.24508370826811282553299619760, 15.9118701853054598411757004472, 17.22536334007634251091044791732, 18.598453461080697301299969681429, 19.19541667698617783967632445187, 20.76900805294698432080241922048, 21.876424793176009860981890495274, 22.44880686505545636907664825494, 23.266865920815464362370554589663, 24.225333660699944214837076295960, 25.02540550317163453216893290250, 26.430659898451308454356124850801, 27.7289317097849874646759511610, 28.56934578370771239462413953997

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