Properties

Label 1-143-143.85-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: $\chi_{143} (85, \cdot )$
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

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

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