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Properties

Label 2-143-11.4-c1-0-8
Degree $2$
Conductor $143$
Sign $0.884 + 0.467i$
Analytic cond. $1.14186$
Root an. cond. $1.06857$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.746 + 0.542i)2-s + (0.321 − 0.990i)3-s + (−0.355 − 1.09i)4-s + (0.296 − 0.215i)5-s + (0.777 − 0.564i)6-s + (−0.326 − 1.00i)7-s + (0.897 − 2.76i)8-s + (1.54 + 1.12i)9-s + 0.338·10-s + (−0.644 + 3.25i)11-s − 1.19·12-s + (0.809 + 0.587i)13-s + (0.300 − 0.926i)14-s + (−0.118 − 0.363i)15-s + (0.308 − 0.224i)16-s + (−4.70 + 3.41i)17-s + ⋯
L(s)  = 1  + (0.527 + 0.383i)2-s + (0.185 − 0.571i)3-s + (−0.177 − 0.546i)4-s + (0.132 − 0.0964i)5-s + (0.317 − 0.230i)6-s + (−0.123 − 0.379i)7-s + (0.317 − 0.976i)8-s + (0.516 + 0.375i)9-s + 0.106·10-s + (−0.194 + 0.980i)11-s − 0.345·12-s + (0.224 + 0.163i)13-s + (0.0804 − 0.247i)14-s + (−0.0304 − 0.0937i)15-s + (0.0771 − 0.0560i)16-s + (−1.14 + 0.828i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 + 0.467i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.884 + 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.884 + 0.467i$
Analytic conductor: \(1.14186\)
Root analytic conductor: \(1.06857\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (92, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 143,\ (\ :1/2),\ 0.884 + 0.467i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41301 - 0.350325i\)
\(L(\frac12)\) \(\approx\) \(1.41301 - 0.350325i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (0.644 - 3.25i)T \)
13 \( 1 + (-0.809 - 0.587i)T \)
good2 \( 1 + (-0.746 - 0.542i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (-0.321 + 0.990i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (-0.296 + 0.215i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (0.326 + 1.00i)T + (-5.66 + 4.11i)T^{2} \)
17 \( 1 + (4.70 - 3.41i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.566 + 1.74i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 1.87T + 23T^{2} \)
29 \( 1 + (-1.41 - 4.35i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.328 + 0.238i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.46 + 7.60i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.000168 - 0.000519i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 11.8T + 43T^{2} \)
47 \( 1 + (-0.814 + 2.50i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-9.92 - 7.21i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.53 + 4.71i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (4.11 - 2.99i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 4.08T + 67T^{2} \)
71 \( 1 + (-3.71 + 2.69i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.826 - 2.54i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-0.564 - 0.410i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-9.56 + 6.95i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 14.3T + 89T^{2} \)
97 \( 1 + (12.1 + 8.85i)T + (29.9 + 92.2i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.23740153565465747818542185147, −12.51816981218162041251112965357, −10.88295373322574904957515907241, −10.03231854439264173253657594378, −8.859714990735048751635635322444, −7.29879367431697320942739451334, −6.68880219270538710401685456882, −5.21615903293357720005209763704, −4.12862484268172288193252508570, −1.75964801105286443457404290785, 2.75314102765545673819014419455, 3.90630579756696903259745252666, 5.08203346810886321393318730773, 6.59385868223479852088800222971, 8.174757244597425506824111295332, 9.031401659756950476247139560639, 10.21647786991535467370775471964, 11.35231772022094088393951106746, 12.17345851397406286397837569231, 13.31534216730225542937100891609

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