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Properties

Label 1-143-143.75-r0-0-0
Degree $1$
Conductor $143$
Sign $0.277 + 0.960i$
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  + (−0.913 + 0.406i)2-s + (0.669 + 0.743i)3-s + (0.669 − 0.743i)4-s + (0.809 − 0.587i)5-s + (−0.913 − 0.406i)6-s + (−0.669 + 0.743i)7-s + (−0.309 + 0.951i)8-s + (−0.104 + 0.994i)9-s + (−0.5 + 0.866i)10-s + 12-s + (0.309 − 0.951i)14-s + (0.978 + 0.207i)15-s + (−0.104 − 0.994i)16-s + (0.913 + 0.406i)17-s + (−0.309 − 0.951i)18-s + (0.978 − 0.207i)19-s + ⋯
L(s)  = 1  + (−0.913 + 0.406i)2-s + (0.669 + 0.743i)3-s + (0.669 − 0.743i)4-s + (0.809 − 0.587i)5-s + (−0.913 − 0.406i)6-s + (−0.669 + 0.743i)7-s + (−0.309 + 0.951i)8-s + (−0.104 + 0.994i)9-s + (−0.5 + 0.866i)10-s + 12-s + (0.309 − 0.951i)14-s + (0.978 + 0.207i)15-s + (−0.104 − 0.994i)16-s + (0.913 + 0.406i)17-s + (−0.309 − 0.951i)18-s + (0.978 − 0.207i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.277 + 0.960i$
Analytic conductor: \(0.664089\)
Root analytic conductor: \(0.664089\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (75, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 143,\ (0:\ ),\ 0.277 + 0.960i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7734560299 + 0.5816469540i\)
\(L(\frac12)\) \(\approx\) \(0.7734560299 + 0.5816469540i\)
\(L(1)\) \(\approx\) \(0.8566124220 + 0.3802808825i\)
\(L(1)\) \(\approx\) \(0.8566124220 + 0.3802808825i\)

Euler product

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

−28.30151774496910144211770315450, −26.61413622912881359790497434504, −26.348658798950029422489673565317, −25.32409232252304007658768973135, −24.6525715169998460132094442601, −23.1512812265979641793970259605, −21.97048682732805404619221280912, −20.671187957687280256701463991561, −20.08990467022694710145169322329, −18.84792395165605756368423148285, −18.3761612773255613553893303363, −17.27070026870268906934955851765, −16.27088233329452878803074163357, −14.68940744830668035225184489330, −13.62706217163996815420950749010, −12.73466795611334620836264244560, −11.45577946539699098453061454753, −10.003363513429710340757859645332, −9.540451173376345981818279706135, −8.02724172663363911994592775553, −7.106655172015880575273878621751, −6.18283408187068803060509364837, −3.51469234854974958917998562155, −2.598066316017325963060774395096, −1.19913635269388515269982447095, 1.77769552624470232221383425014, 3.12643168507196764719539970518, 5.16103319580872286889119253764, 6.02854204499571178529001523091, 7.720777294466040814311237294284, 8.84486887966424626274515825858, 9.57522394279348078000105648287, 10.23644206644807177445989013142, 11.84887073741313251886225784865, 13.38508122669743531172981761359, 14.48867896943153835287535234367, 15.598212854025226017827357798047, 16.3103952270835253668741983870, 17.27128194892023173836538232983, 18.51078059168226174866721266041, 19.4839200116851453384915322545, 20.433618138330379915029370863119, 21.32665856891749372407445180131, 22.33994348863879594788154626475, 23.99185838997360314592082992652, 25.08578500991105413631848297158, 25.55061341129095114913579710357, 26.35897160806703960682793757506, 27.533019695351646650588515897542, 28.326800620585886521516716793156

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