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Properties

Label 2-143-13.3-c3-0-11
Degree $2$
Conductor $143$
Sign $-0.641 + 0.766i$
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  + (−2.55 − 4.42i)2-s + (1.79 + 3.10i)3-s + (−9.08 + 15.7i)4-s − 2.16·5-s + (9.16 − 15.8i)6-s + (−10.4 + 18.1i)7-s + 51.9·8-s + (7.08 − 12.2i)9-s + (5.54 + 9.60i)10-s + (−5.5 − 9.52i)11-s − 65.0·12-s + (−3.83 − 46.7i)13-s + 107.·14-s + (−3.88 − 6.72i)15-s + (−60.3 − 104. i)16-s + (50.3 − 87.2i)17-s + ⋯
L(s)  = 1  + (−0.904 − 1.56i)2-s + (0.344 + 0.596i)3-s + (−1.13 + 1.96i)4-s − 0.193·5-s + (0.623 − 1.07i)6-s + (−0.566 + 0.981i)7-s + 2.29·8-s + (0.262 − 0.454i)9-s + (0.175 + 0.303i)10-s + (−0.150 − 0.261i)11-s − 1.56·12-s + (−0.0819 − 0.996i)13-s + 2.04·14-s + (−0.0668 − 0.115i)15-s + (−0.942 − 1.63i)16-s + (0.718 − 1.24i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.641 + 0.766i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.641 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $-0.641 + 0.766i$
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: \((2,\ 143,\ (\ :3/2),\ -0.641 + 0.766i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.358223 - 0.767058i\)
\(L(\frac12)\) \(\approx\) \(0.358223 - 0.767058i\)
\(L(\frac{5}{2})\) 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 \( 1 + (3.83 + 46.7i)T \)
good2 \( 1 + (2.55 + 4.42i)T + (-4 + 6.92i)T^{2} \)
3 \( 1 + (-1.79 - 3.10i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + 2.16T + 125T^{2} \)
7 \( 1 + (10.4 - 18.1i)T + (-171.5 - 297. i)T^{2} \)
17 \( 1 + (-50.3 + 87.2i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-81.9 + 142. i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-23.7 - 41.1i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-122. - 212. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 29.8T + 2.97e4T^{2} \)
37 \( 1 + (97.8 + 169. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (98.5 + 170. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-4.53 + 7.85i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 449.T + 1.03e5T^{2} \)
53 \( 1 + 296.T + 1.48e5T^{2} \)
59 \( 1 + (-353. + 613. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-126. + 219. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (220. + 381. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (169. - 293. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 188.T + 3.89e5T^{2} \)
79 \( 1 - 252.T + 4.93e5T^{2} \)
83 \( 1 + 710.T + 5.71e5T^{2} \)
89 \( 1 + (691. + 1.19e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-186. + 323. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−12.09515091512761692456813943714, −11.18070156630948920159481588577, −10.06674043744662911280112834354, −9.343622955574489467332581427850, −8.763027821067128226549350265498, −7.35947857366398805915528729732, −5.17707904939319085685846952366, −3.40207566236443705095428379845, −2.78562067470661943010272232632, −0.61879256005014365875930000509, 1.34122336151489125939415100643, 4.16509445911691848025884486745, 5.85646415265252049996162456663, 6.90136644095777463685982417295, 7.69480626394365559222247861270, 8.322273259082137673903974449042, 9.836529616310758528104057777201, 10.28644015229205795150288174244, 12.17526658621775461551924754033, 13.55586291473185593700420337259

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