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Properties

Label 2-143-13.3-c3-0-10
Degree $2$
Conductor $143$
Sign $-0.207 - 0.978i$
Analytic cond. $8.43727$
Root an. cond. $2.90469$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.534 + 0.925i)2-s + (4.92 + 8.53i)3-s + (3.42 − 5.93i)4-s + 2.35·5-s + (−5.26 + 9.11i)6-s + (−6.12 + 10.6i)7-s + 15.8·8-s + (−35.0 + 60.6i)9-s + (1.26 + 2.18i)10-s + (−5.5 − 9.52i)11-s + 67.5·12-s + (39.1 + 25.8i)13-s − 13.0·14-s + (11.6 + 20.1i)15-s + (−18.9 − 32.8i)16-s + (10.6 − 18.3i)17-s + ⋯
L(s)  = 1  + (0.188 + 0.327i)2-s + (0.947 + 1.64i)3-s + (0.428 − 0.742i)4-s + 0.211·5-s + (−0.358 + 0.620i)6-s + (−0.330 + 0.572i)7-s + 0.701·8-s + (−1.29 + 2.24i)9-s + (0.0398 + 0.0690i)10-s + (−0.150 − 0.261i)11-s + 1.62·12-s + (0.834 + 0.550i)13-s − 0.249·14-s + (0.200 + 0.346i)15-s + (−0.296 − 0.512i)16-s + (0.151 − 0.262i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.207 - 0.978i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.207 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $-0.207 - 0.978i$
Analytic conductor: \(8.43727\)
Root analytic conductor: \(2.90469\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 143,\ (\ :3/2),\ -0.207 - 0.978i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.70228 + 2.10071i\)
\(L(\frac12)\) \(\approx\) \(1.70228 + 2.10071i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (5.5 + 9.52i)T \)
13 \( 1 + (-39.1 - 25.8i)T \)
good2 \( 1 + (-0.534 - 0.925i)T + (-4 + 6.92i)T^{2} \)
3 \( 1 + (-4.92 - 8.53i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 - 2.35T + 125T^{2} \)
7 \( 1 + (6.12 - 10.6i)T + (-171.5 - 297. i)T^{2} \)
17 \( 1 + (-10.6 + 18.3i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-16.6 + 28.8i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-55.6 - 96.3i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (106. + 184. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 177.T + 2.97e4T^{2} \)
37 \( 1 + (90.8 + 157. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-90.7 - 157. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-182. + 315. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 180.T + 1.03e5T^{2} \)
53 \( 1 + 290.T + 1.48e5T^{2} \)
59 \( 1 + (-178. + 309. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-285. + 494. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (541. + 937. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (440. - 762. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 709.T + 3.89e5T^{2} \)
79 \( 1 + 93.0T + 4.93e5T^{2} \)
83 \( 1 - 1.23e3T + 5.71e5T^{2} \)
89 \( 1 + (-494. - 856. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-54.0 + 93.6i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−13.58803323454234461074954646944, −11.50613627744646343001592451223, −10.71921333187803169860063498530, −9.640664514631683722832632581340, −9.159302022594031927405261002264, −7.82532703057278749222117515741, −6.05806563776127659296418150228, −5.11271986900106159670269795476, −3.79104318012574978033839788361, −2.36265847331229866754912426207, 1.29214836030696106019156352119, 2.65371485451897655869927768389, 3.69017352423209636048971603590, 6.20146201718516044429397455029, 7.17946310072567680650698826103, 7.934419243102780114484832389521, 8.842488751594655217740392972471, 10.45822006349141989938005524886, 11.75944486159101748419059739587, 12.63398347097438651309625145686

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