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Properties

Label 2-143-11.4-c1-0-0
Degree $2$
Conductor $143$
Sign $-0.190 - 0.981i$
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  + (−0.743 − 0.540i)2-s + (−0.330 + 1.01i)3-s + (−0.356 − 1.09i)4-s + (−2.64 + 1.92i)5-s + (0.794 − 0.577i)6-s + (0.656 + 2.02i)7-s + (−0.896 + 2.75i)8-s + (1.50 + 1.09i)9-s + 3.00·10-s + (−2.93 + 1.53i)11-s + 1.23·12-s + (−0.809 − 0.587i)13-s + (0.603 − 1.85i)14-s + (−1.07 − 3.32i)15-s + (0.289 − 0.210i)16-s + (−2.27 + 1.65i)17-s + ⋯
L(s)  = 1  + (−0.526 − 0.382i)2-s + (−0.190 + 0.586i)3-s + (−0.178 − 0.548i)4-s + (−1.18 + 0.859i)5-s + (0.324 − 0.235i)6-s + (0.248 + 0.763i)7-s + (−0.316 + 0.975i)8-s + (0.501 + 0.364i)9-s + 0.950·10-s + (−0.885 + 0.463i)11-s + 0.356·12-s + (−0.224 − 0.163i)13-s + (0.161 − 0.496i)14-s + (−0.278 − 0.857i)15-s + (0.0724 − 0.0526i)16-s + (−0.552 + 0.401i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.190 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.190 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $-0.190 - 0.981i$
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: \((2,\ 143,\ (\ :1/2),\ -0.190 - 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.301862 + 0.366184i\)
\(L(\frac12)\) \(\approx\) \(0.301862 + 0.366184i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (2.93 - 1.53i)T \)
13 \( 1 + (0.809 + 0.587i)T \)
good2 \( 1 + (0.743 + 0.540i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (0.330 - 1.01i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (2.64 - 1.92i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (-0.656 - 2.02i)T + (-5.66 + 4.11i)T^{2} \)
17 \( 1 + (2.27 - 1.65i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.205 + 0.631i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 4.97T + 23T^{2} \)
29 \( 1 + (1.48 + 4.57i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.969 + 0.704i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-1.42 - 4.39i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.32 + 7.16i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 0.661T + 43T^{2} \)
47 \( 1 + (2.37 - 7.29i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-6.50 - 4.72i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-3.17 - 9.77i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (5.70 - 4.14i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 14.7T + 67T^{2} \)
71 \( 1 + (11.6 - 8.49i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (0.428 + 1.31i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-10.8 - 7.89i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-2.85 + 2.07i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 8.08T + 89T^{2} \)
97 \( 1 + (5.75 + 4.18i)T + (29.9 + 92.2i)T^{2} \)
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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

−13.33456353987279616812175441909, −12.00199759681544078557145278494, −11.01926867022614573193915050016, −10.56124066621733421030101256701, −9.491860857669546383841494978630, −8.277888262637818326679979415168, −7.21372600302738311754617739628, −5.49808762540636161759999048541, −4.36774669591879992557782826444, −2.51818931983750898124792991185, 0.58431306692851016343287841099, 3.65595861188290435792026527542, 4.81543340700586020028747046416, 6.87567044416068989540123652386, 7.58713010791870398903659972108, 8.350675975714454604646060931290, 9.410191467901515417101701312239, 10.94419937656920311807119749680, 12.00287869357938516091735671667, 12.82750672606205027990888685547

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