Properties

Label 1-143-143.63-r0-0-0
Degree $1$
Conductor $143$
Sign $-0.999 + 0.00342i$
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  + (0.743 − 0.669i)2-s + (−0.104 − 0.994i)3-s + (0.104 − 0.994i)4-s + (−0.951 + 0.309i)5-s + (−0.743 − 0.669i)6-s + (−0.994 − 0.104i)7-s + (−0.587 − 0.809i)8-s + (−0.978 + 0.207i)9-s + (−0.5 + 0.866i)10-s − 12-s + (−0.809 + 0.587i)14-s + (0.406 + 0.913i)15-s + (−0.978 − 0.207i)16-s + (0.669 − 0.743i)17-s + (−0.587 + 0.809i)18-s + (0.406 − 0.913i)19-s + ⋯
L(s)  = 1  + (0.743 − 0.669i)2-s + (−0.104 − 0.994i)3-s + (0.104 − 0.994i)4-s + (−0.951 + 0.309i)5-s + (−0.743 − 0.669i)6-s + (−0.994 − 0.104i)7-s + (−0.587 − 0.809i)8-s + (−0.978 + 0.207i)9-s + (−0.5 + 0.866i)10-s − 12-s + (−0.809 + 0.587i)14-s + (0.406 + 0.913i)15-s + (−0.978 − 0.207i)16-s + (0.669 − 0.743i)17-s + (−0.587 + 0.809i)18-s + (0.406 − 0.913i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $-0.999 + 0.00342i$
Analytic conductor: \(0.664089\)
Root analytic conductor: \(0.664089\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 143,\ (0:\ ),\ -0.999 + 0.00342i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.001639347303 - 0.9565729093i\)
\(L(\frac12)\) \(\approx\) \(0.001639347303 - 0.9565729093i\)
\(L(1)\) \(\approx\) \(0.6594003007 - 0.8011081680i\)
\(L(1)\) \(\approx\) \(0.6594003007 - 0.8011081680i\)

Euler product

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

−28.666538150707835898522357432041, −27.61962134651934963480575863035, −26.66744156899574726873081532024, −25.890247054874402964400889949364, −24.8775485033277425497585219642, −23.52063133097561324765635508261, −22.92927602231507150816574679051, −22.09262197535635264824192126980, −21.025270859888364724732045887501, −20.11448797035437888587282232054, −18.91271692799615134100003929808, −17.0316676500884057682449961842, −16.49266033131012452687485054200, −15.50472109171192480357857421620, −14.98027547477550085628713882436, −13.56786462948177421188784525083, −12.34080051191676469835143614855, −11.53241760764039747576458021838, −10.01563044007222079062962935065, −8.75096355816263282304724742630, −7.67022638264980998536314269258, −6.20044538667062027911956163646, −5.13500127036628115423263388543, −3.84832659012374216554679429112, −3.245984150303625626476983563372, 0.66800215537234075228704215257, 2.619730655802812866010280264990, 3.511504724463247280123536455441, 5.135712636180915082058977596875, 6.554004443653284207813190181983, 7.303387487932824385056187755055, 8.9814024266695302841302905110, 10.48260674255451353050549126174, 11.59014060083879927960608964044, 12.33888909176450540059280611848, 13.25238216969035050757580733645, 14.245636128173166845151893042194, 15.42340483899733875255491477644, 16.53203088142569088332477135955, 18.26055303330765579064885891318, 19.0429525743737294758640379863, 19.69684769005065841912675129540, 20.6038422727630048282924731005, 22.31593665219344537804360135692, 22.79741908945885435817714185359, 23.63851400634842345441678075407, 24.50768476635599588956156134031, 25.66140831613982571810471506927, 26.95160757563986310408810449314, 28.2628910047775026566126496362

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