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Properties

Label 2-143-13.3-c1-0-4
Degree 22
Conductor 143143
Sign 0.999+0.0260i0.999 + 0.0260i
Analytic cond. 1.141861.14186
Root an. cond. 1.068571.06857
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(143s/2 ΓC(s) L(s)=((0.999+0.0260i) Λ‾(2−s)\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}
Λ(s)=(143s/2 ΓC(s+1/2) L(s)=((0.999+0.0260i) Λ‾(1−s)\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: 22
Conductor: 143143    =    11⋅1311 \cdot 13
Sign: 0.999+0.0260i0.999 + 0.0260i
Analytic conductor: 1.141861.14186
Root analytic conductor: 1.068571.06857
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ143(133,⋅)\chi_{143} (133, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 143, ( :1/2), 0.999+0.0260i)(2,\ 143,\ (\ :1/2),\ 0.999 + 0.0260i)

Particular Values

L(1)L(1) ≈\approx 1.24720−0.0162425i1.24720 - 0.0162425i
L(12)L(\frac12) ≈\approx 1.24720−0.0162425i1.24720 - 0.0162425i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad11 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
13 1+(−2.63+2.46i)T 1 + (-2.63 + 2.46i)T
good2 1+(0.5+0.866i)T+(−1+1.73i)T2 1 + (0.5 + 0.866i)T + (-1 + 1.73i)T^{2}
3 1+(−1.13−1.95i)T+(−1.5+2.59i)T2 1 + (-1.13 - 1.95i)T + (-1.5 + 2.59i)T^{2}
5 1−2.11T+5T2 1 - 2.11T + 5T^{2}
7 1+(1.68−2.92i)T+(−3.5−6.06i)T2 1 + (1.68 - 2.92i)T + (-3.5 - 6.06i)T^{2}
17 1+(−1.07+1.85i)T+(−8.5−14.7i)T2 1 + (-1.07 + 1.85i)T + (-8.5 - 14.7i)T^{2}
19 1+(1.57−2.72i)T+(−9.5−16.4i)T2 1 + (1.57 - 2.72i)T + (-9.5 - 16.4i)T^{2}
23 1+(2.26+3.91i)T+(−11.5+19.9i)T2 1 + (2.26 + 3.91i)T + (-11.5 + 19.9i)T^{2}
29 1+(1.74+3.02i)T+(−14.5+25.1i)T2 1 + (1.74 + 3.02i)T + (-14.5 + 25.1i)T^{2}
31 1+9.27T+31T2 1 + 9.27T + 31T^{2}
37 1+(−4.58−7.93i)T+(−18.5+32.0i)T2 1 + (-4.58 - 7.93i)T + (-18.5 + 32.0i)T^{2}
41 1+(5.00+8.67i)T+(−20.5+35.5i)T2 1 + (5.00 + 8.67i)T + (-20.5 + 35.5i)T^{2}
43 1+(1.26−2.18i)T+(−21.5−37.2i)T2 1 + (1.26 - 2.18i)T + (-21.5 - 37.2i)T^{2}
47 1+1.96T+47T2 1 + 1.96T + 47T^{2}
53 1−7.75T+53T2 1 - 7.75T + 53T^{2}
59 1+(−3.01+5.22i)T+(−29.5−51.0i)T2 1 + (-3.01 + 5.22i)T + (-29.5 - 51.0i)T^{2}
61 1+(4.89−8.47i)T+(−30.5−52.8i)T2 1 + (4.89 - 8.47i)T + (-30.5 - 52.8i)T^{2}
67 1+(2.01+3.49i)T+(−33.5+58.0i)T2 1 + (2.01 + 3.49i)T + (-33.5 + 58.0i)T^{2}
71 1+(3−5.19i)T+(−35.5−61.4i)T2 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2}
73 1−10.8T+73T2 1 - 10.8T + 73T^{2}
79 1−1.66T+79T2 1 - 1.66T + 79T^{2}
83 1−5.66T+83T2 1 - 5.66T + 83T^{2}
89 1+(4.67+8.09i)T+(−44.5+77.0i)T2 1 + (4.67 + 8.09i)T + (-44.5 + 77.0i)T^{2}
97 1+(5.81−10.0i)T+(−48.5−84.0i)T2 1 + (5.81 - 10.0i)T + (-48.5 - 84.0i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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 ZZ-function along the critical line