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Properties

Label 1-143-143.49-r0-0-0
Degree $1$
Conductor $143$
Sign $-0.252 + 0.967i$
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  + (0.104 + 0.994i)2-s + (−0.978 − 0.207i)3-s + (−0.978 + 0.207i)4-s + (0.809 + 0.587i)5-s + (0.104 − 0.994i)6-s + (0.978 − 0.207i)7-s + (−0.309 − 0.951i)8-s + (0.913 + 0.406i)9-s + (−0.5 + 0.866i)10-s + 12-s + (0.309 + 0.951i)14-s + (−0.669 − 0.743i)15-s + (0.913 − 0.406i)16-s + (−0.104 + 0.994i)17-s + (−0.309 + 0.951i)18-s + (−0.669 + 0.743i)19-s + ⋯
L(s)  = 1  + (0.104 + 0.994i)2-s + (−0.978 − 0.207i)3-s + (−0.978 + 0.207i)4-s + (0.809 + 0.587i)5-s + (0.104 − 0.994i)6-s + (0.978 − 0.207i)7-s + (−0.309 − 0.951i)8-s + (0.913 + 0.406i)9-s + (−0.5 + 0.866i)10-s + 12-s + (0.309 + 0.951i)14-s + (−0.669 − 0.743i)15-s + (0.913 − 0.406i)16-s + (−0.104 + 0.994i)17-s + (−0.309 + 0.951i)18-s + (−0.669 + 0.743i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $-0.252 + 0.967i$
Analytic conductor: \(0.664089\)
Root analytic conductor: \(0.664089\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 143,\ (0:\ ),\ -0.252 + 0.967i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5691079438 + 0.7369166103i\)
\(L(\frac12)\) \(\approx\) \(0.5691079438 + 0.7369166103i\)
\(L(1)\) \(\approx\) \(0.7661989646 + 0.5198288829i\)
\(L(1)\) \(\approx\) \(0.7661989646 + 0.5198288829i\)

Euler product

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

−28.16558955337642723601442658934, −27.50152174020974378942173959088, −26.43807904280269310750293601849, −24.74567492244536179654008603933, −23.92559567634103938622822346719, −22.82601134289350634901358156039, −21.8353630488626008946246301340, −21.13031903855930929638723194867, −20.43759968721808599394171469720, −18.88592875828923565436316113564, −17.69439413284787347570724226891, −17.489763032013477698102176171185, −16.02617124633185625656735848774, −14.49820412820255943434887865759, −13.40642313089611368706022076440, −12.33613319672356453075281904918, −11.49676837386717751036940987878, −10.49276170418230432647116897114, −9.51094387257429679473194104681, −8.35662730609498310723167662296, −6.312727094601765455757777216497, −5.019899181795441757389828690809, −4.53000643699899567687106129005, −2.38333172475576454071575711523, −1.035562026678719863519386435121, 1.65484880343240132011446160135, 4.069129608357824342406666199820, 5.34382341723689776335939757458, 6.14772025552932297531632986356, 7.181630155596986032051033483441, 8.33002529418411642870332374105, 9.95490389098734731088357564588, 10.85028029890581072299036115175, 12.27900940518078848804239297374, 13.445635942029014293449463079384, 14.38599647469148513964246650568, 15.391234618969345194010931362627, 16.74107894560326807205817337540, 17.4853616260435802420134108757, 18.04596751482083657754505516259, 19.09419363784181678121191112286, 21.23342631782756266060321545758, 21.78078684688259193473158931425, 22.87926834218391814801972222020, 23.687545204408530903008104468067, 24.53678465913202146882978826203, 25.49422155940059697434006565419, 26.581619270047721291829267400611, 27.51381005152750504245286610133, 28.37264234122918082147527827720

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