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Properties

Label 2-143-13.3-c1-0-4
Degree $2$
Conductor $143$
Sign $0.999 + 0.0260i$
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  + (−0.5 − 0.866i)2-s + (1.13 + 1.95i)3-s + (0.500 − 0.866i)4-s + 2.11·5-s + (1.13 − 1.95i)6-s + (−1.68 + 2.92i)7-s − 3·8-s + (−1.05 + 1.83i)9-s + (−1.05 − 1.83i)10-s + (−0.5 − 0.866i)11-s + 2.26·12-s + (2.63 − 2.46i)13-s + 3.37·14-s + (2.39 + 4.14i)15-s + (0.500 + 0.866i)16-s + (1.07 − 1.85i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.652 + 1.13i)3-s + (0.250 − 0.433i)4-s + 0.946·5-s + (0.461 − 0.799i)6-s + (−0.638 + 1.10i)7-s − 1.06·8-s + (−0.352 + 0.610i)9-s + (−0.334 − 0.579i)10-s + (−0.150 − 0.261i)11-s + 0.652·12-s + (0.729 − 0.683i)13-s + 0.902·14-s + (0.617 + 1.07i)15-s + (0.125 + 0.216i)16-s + (0.260 − 0.450i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0260i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0260i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.999 + 0.0260i$
Analytic conductor: \(1.14186\)
Root analytic conductor: \(1.06857\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 143,\ (\ :1/2),\ 0.999 + 0.0260i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24720 - 0.0162425i\)
\(L(\frac12)\) \(\approx\) \(1.24720 - 0.0162425i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-2.63 + 2.46i)T \)
good2 \( 1 + (0.5 + 0.866i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-1.13 - 1.95i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 2.11T + 5T^{2} \)
7 \( 1 + (1.68 - 2.92i)T + (-3.5 - 6.06i)T^{2} \)
17 \( 1 + (-1.07 + 1.85i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.57 - 2.72i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.26 + 3.91i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.74 + 3.02i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 9.27T + 31T^{2} \)
37 \( 1 + (-4.58 - 7.93i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.00 + 8.67i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.26 - 2.18i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 1.96T + 47T^{2} \)
53 \( 1 - 7.75T + 53T^{2} \)
59 \( 1 + (-3.01 + 5.22i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.89 - 8.47i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.01 + 3.49i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 10.8T + 73T^{2} \)
79 \( 1 - 1.66T + 79T^{2} \)
83 \( 1 - 5.66T + 83T^{2} \)
89 \( 1 + (4.67 + 8.09i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.81 - 10.0i)T + (-48.5 - 84.0i)T^{2} \)
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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

−13.07728644627150474865954980049, −11.94502336539938344143777531956, −10.64326736106762772841209177549, −9.978352392356170454569750475560, −9.295529600099973644716534589588, −8.515584839735123468232951653704, −6.20895283757061120201073575440, −5.47163819326285002074482755230, −3.43047573360283080482015223790, −2.29811509604988134280733902110, 1.93613738260217822756478514708, 3.58607817370604768313869934116, 6.02524244471339962690809683317, 6.93691400583185174337922837809, 7.57231993618432392855634410253, 8.740898625700242982666304028379, 9.713359903749772567673275696364, 11.08970174697276790983912334354, 12.55959240627443526701704241307, 13.25530523283608866065164338213

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