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Properties

Label 1-143-143.106-r0-0-0
Degree $1$
Conductor $143$
Sign $-0.861 - 0.508i$
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  + (0.207 − 0.978i)2-s + (0.913 − 0.406i)3-s + (−0.913 − 0.406i)4-s + (−0.951 − 0.309i)5-s + (−0.207 − 0.978i)6-s + (0.406 − 0.913i)7-s + (−0.587 + 0.809i)8-s + (0.669 − 0.743i)9-s + (−0.5 + 0.866i)10-s − 12-s + (−0.809 − 0.587i)14-s + (−0.994 + 0.104i)15-s + (0.669 + 0.743i)16-s + (−0.978 + 0.207i)17-s + (−0.587 − 0.809i)18-s + (−0.994 − 0.104i)19-s + ⋯
L(s)  = 1  + (0.207 − 0.978i)2-s + (0.913 − 0.406i)3-s + (−0.913 − 0.406i)4-s + (−0.951 − 0.309i)5-s + (−0.207 − 0.978i)6-s + (0.406 − 0.913i)7-s + (−0.587 + 0.809i)8-s + (0.669 − 0.743i)9-s + (−0.5 + 0.866i)10-s − 12-s + (−0.809 − 0.587i)14-s + (−0.994 + 0.104i)15-s + (0.669 + 0.743i)16-s + (−0.978 + 0.207i)17-s + (−0.587 − 0.809i)18-s + (−0.994 − 0.104i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.861 - 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.861 - 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $-0.861 - 0.508i$
Analytic conductor: \(0.664089\)
Root analytic conductor: \(0.664089\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 143,\ (0:\ ),\ -0.861 - 0.508i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3228182441 - 1.182523608i\)
\(L(\frac12)\) \(\approx\) \(0.3228182441 - 1.182523608i\)
\(L(1)\) \(\approx\) \(0.7972814607 - 0.8934270376i\)
\(L(1)\) \(\approx\) \(0.7972814607 - 0.8934270376i\)

Euler product

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

−28.20945786717400564937711191784, −27.31497986143402059332475892373, −26.72532158230233494297522858135, −25.67875835557933483441197024487, −24.851849546498410907935873730812, −24.00153226402228921724021015733, −22.84776492735188374146150048390, −21.85016736710266213073937184818, −20.97886849057265217437954402468, −19.48425039073572567374972304925, −18.783895221228234431158195006462, −17.580669756359862687442549143450, −16.109220641319033714738601619073, −15.320804921610138929445804346191, −14.88242648506303108823263193931, −13.70260459328740125440381283404, −12.528991658908926656909557870906, −11.16329222805414117296029886145, −9.50490472381447759662036236943, −8.49358715085389682373623025605, −7.83228827468027662267825443853, −6.53932606879591163934341635431, −4.90345999331595737608653496870, −3.969316265437746163770731096343, −2.62415584757463229411418970155, 0.99915284017337753495960729594, 2.521143183561183900901711330914, 3.90723810470799199323307424066, 4.56879731552510740040735023154, 6.83211748586797273810016404078, 8.169395090249685607064657168253, 8.87694942271788454644942512375, 10.36764164040630111976887337556, 11.365240869771977533960430991039, 12.601284777560564768827595462921, 13.323207834179063030832074972899, 14.45422381877658338633438346173, 15.2915878446537374591811854939, 16.96797338126578341800207640454, 18.24403983250910735682516100158, 19.2370883827076780764595294124, 20.026995134836453379326734437286, 20.52774178686554297096031436867, 21.65780099601985012813183855547, 23.15000178541781864924052793113, 23.74046175488362214356042609670, 24.69206370730285715229826268461, 26.36944791191569053259336236285, 26.89412295893091998390288484557, 27.865657638678983637427250315755

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