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Properties

Label 1-143-143.119-r1-0-0
Degree $1$
Conductor $143$
Sign $0.289 + 0.957i$
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  + (−0.406 − 0.913i)2-s + (0.669 + 0.743i)3-s + (−0.669 + 0.743i)4-s + (0.587 + 0.809i)5-s + (0.406 − 0.913i)6-s + (0.743 + 0.669i)7-s + (0.951 + 0.309i)8-s + (−0.104 + 0.994i)9-s + (0.5 − 0.866i)10-s − 12-s + (0.309 − 0.951i)14-s + (−0.207 + 0.978i)15-s + (−0.104 − 0.994i)16-s + (−0.913 − 0.406i)17-s + (0.951 − 0.309i)18-s + (0.207 + 0.978i)19-s + ⋯
L(s)  = 1  + (−0.406 − 0.913i)2-s + (0.669 + 0.743i)3-s + (−0.669 + 0.743i)4-s + (0.587 + 0.809i)5-s + (0.406 − 0.913i)6-s + (0.743 + 0.669i)7-s + (0.951 + 0.309i)8-s + (−0.104 + 0.994i)9-s + (0.5 − 0.866i)10-s − 12-s + (0.309 − 0.951i)14-s + (−0.207 + 0.978i)15-s + (−0.104 − 0.994i)16-s + (−0.913 − 0.406i)17-s + (0.951 − 0.309i)18-s + (0.207 + 0.978i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.289 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.289 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.289 + 0.957i$
Analytic conductor: \(15.3674\)
Root analytic conductor: \(15.3674\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 143,\ (1:\ ),\ 0.289 + 0.957i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.489034510 + 1.104729436i\)
\(L(\frac12)\) \(\approx\) \(1.489034510 + 1.104729436i\)
\(L(1)\) \(\approx\) \(1.173620899 + 0.2456673188i\)
\(L(1)\) \(\approx\) \(1.173620899 + 0.2456673188i\)

Euler product

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

−27.705702196727918967554860027895, −26.57413190380178907884967910472, −25.78876879430436636045722010142, −24.84662561356218945518482257077, −24.0618899337415540391464802183, −23.62263859998771767256948811910, −21.95079676465302667192245071113, −20.51776906766189015221427177460, −19.85654013633598205626167146678, −18.61322625663265377906519386085, −17.516215165156846003102158585921, −17.135368524263705687466042228838, −15.60956253287708138877851408871, −14.56201547734923097405938218968, −13.543482961689089755828022706666, −13.02458525778752509540778007563, −11.17571873634948670145534721984, −9.57885366883675462445643628215, −8.77913714541632613393761546687, −7.78687153521270370685617248139, −6.817295487955588273646006767409, −5.48513477291720974916444813923, −4.21157194159212596612792036257, −1.9042544415229415416365353689, −0.77677006441035665573504506812, 1.9399938416140803167806142451, 2.76770860743130764595770617898, 4.1002109482896685641984148300, 5.44065312227508532764078190010, 7.466379371687210581306178139024, 8.70057029142545193663043929914, 9.48912044936930932850712512494, 10.630249185421278319900492822281, 11.29388385342197252110940711154, 12.82052172000787654431146016818, 14.09811035334036084227680308517, 14.72487193783433901016421658302, 16.13878506220384324766002844870, 17.4957373156319082789910683493, 18.39501379714107060509592435923, 19.23258188518076373861842017771, 20.53511834170593621889332455181, 21.12093102097760898453559357181, 22.04680665777292596035209027687, 22.68747340823137910415704634053, 24.72302937198305687248611341284, 25.56194117979493559837814584550, 26.62787359110464049941131210193, 27.126239858329682414080513934080, 28.207081684834503290133017795287

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