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Properties

Label 1-143-143.7-r0-0-0
Degree $1$
Conductor $143$
Sign $0.833 + 0.551i$
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  + (−0.743 − 0.669i)2-s + (−0.104 + 0.994i)3-s + (0.104 + 0.994i)4-s + (0.951 + 0.309i)5-s + (0.743 − 0.669i)6-s + (0.994 − 0.104i)7-s + (0.587 − 0.809i)8-s + (−0.978 − 0.207i)9-s + (−0.5 − 0.866i)10-s − 12-s + (−0.809 − 0.587i)14-s + (−0.406 + 0.913i)15-s + (−0.978 + 0.207i)16-s + (0.669 + 0.743i)17-s + (0.587 + 0.809i)18-s + (−0.406 − 0.913i)19-s + ⋯
L(s)  = 1  + (−0.743 − 0.669i)2-s + (−0.104 + 0.994i)3-s + (0.104 + 0.994i)4-s + (0.951 + 0.309i)5-s + (0.743 − 0.669i)6-s + (0.994 − 0.104i)7-s + (0.587 − 0.809i)8-s + (−0.978 − 0.207i)9-s + (−0.5 − 0.866i)10-s − 12-s + (−0.809 − 0.587i)14-s + (−0.406 + 0.913i)15-s + (−0.978 + 0.207i)16-s + (0.669 + 0.743i)17-s + (0.587 + 0.809i)18-s + (−0.406 − 0.913i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.833 + 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.833 + 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.833 + 0.551i$
Analytic conductor: \(0.664089\)
Root analytic conductor: \(0.664089\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 143,\ (0:\ ),\ 0.833 + 0.551i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8766079892 + 0.2637763668i\)
\(L(\frac12)\) \(\approx\) \(0.8766079892 + 0.2637763668i\)
\(L(1)\) \(\approx\) \(0.8825372135 + 0.1160116373i\)
\(L(1)\) \(\approx\) \(0.8825372135 + 0.1160116373i\)

Euler product

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

−28.15222276290141878711468060143, −27.30547024527525908534675919510, −25.935460420396653746172675455103, −25.12500179351459219900242510952, −24.526425349115022364705187818122, −23.700694145027805020649109649669, −22.58751423447173519885021148470, −20.96018849440485087883834083833, −20.12649417649757013368171128746, −18.527420261110101771822804917299, −18.34458776986875095185878404142, −17.11898562925582544258564773450, −16.61075039435642587190234596603, −14.728588787023337383251433964620, −14.157824592514787280794596905604, −12.95994242141706174132389837351, −11.59304490752666338858938785389, −10.41277159186000547067246220498, −9.06364468209146550006400269368, −8.148652793604465932190171176462, −7.10617164278159807029129240444, −5.895774455771277304126222363031, −5.092538290966970517489934369186, −2.22163765692337832728631307696, −1.23306566585621173014053020085, 1.68560080491862384345333229309, 3.06812207846606327672037713149, 4.46053038207032462627506660470, 5.80331520194776495455056968828, 7.54578831414371802410182117350, 8.88105141644993981823046844090, 9.669283177282373776403888347991, 10.79758784481136041540717346349, 11.27497939072077679085857498171, 12.86387909261407860706472233994, 14.19277422626013578363873222038, 15.212286353877362067263563703889, 16.686645822121247441984646646463, 17.37284360047381814808864586769, 18.13875042118216556818228475428, 19.51780163781647301723071754400, 20.632230734538953080386841119851, 21.44522022974289736857080822017, 21.80071013029305654479127141277, 23.1987629199744577624764224979, 24.80857566553370399985384853140, 25.90830543882107173270067537079, 26.45737064212606567524269008559, 27.63796556630704528166197558997, 28.1132557472110240280777773894

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