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Properties

Label 2-143-13.9-c3-0-2
Degree $2$
Conductor $143$
Sign $0.305 - 0.952i$
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 + (−2.28 + 3.96i)3-s + (−9.03 − 15.6i)4-s − 15.4·5-s + (11.6 + 20.2i)6-s + (8.38 + 14.5i)7-s − 51.4·8-s + (3.03 + 5.25i)9-s + (−39.4 + 68.2i)10-s + (−5.5 + 9.52i)11-s + 82.6·12-s + (−41.5 + 21.7i)13-s + 85.5·14-s + (35.3 − 61.1i)15-s + (−58.9 + 102. i)16-s + (7.01 + 12.1i)17-s + ⋯
L(s)  = 1  + (0.902 − 1.56i)2-s + (−0.440 + 0.762i)3-s + (−1.12 − 1.95i)4-s − 1.38·5-s + (0.794 + 1.37i)6-s + (0.452 + 0.783i)7-s − 2.27·8-s + (0.112 + 0.194i)9-s + (−1.24 + 2.15i)10-s + (−0.150 + 0.261i)11-s + 1.98·12-s + (−0.886 + 0.463i)13-s + 1.63·14-s + (0.607 − 1.05i)15-s + (−0.921 + 1.59i)16-s + (0.100 + 0.173i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.305 - 0.952i$
Analytic conductor: \(8.43727\)
Root analytic conductor: \(2.90469\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 143,\ (\ :3/2),\ 0.305 - 0.952i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.392990 + 0.286708i\)
\(L(\frac12)\) \(\approx\) \(0.392990 + 0.286708i\)
\(L(\frac{5}{2})\) 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 \( 1 + (41.5 - 21.7i)T \)
good2 \( 1 + (-2.55 + 4.42i)T + (-4 - 6.92i)T^{2} \)
3 \( 1 + (2.28 - 3.96i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + 15.4T + 125T^{2} \)
7 \( 1 + (-8.38 - 14.5i)T + (-171.5 + 297. i)T^{2} \)
17 \( 1 + (-7.01 - 12.1i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (37.7 + 65.3i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (92.5 - 160. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-66.6 + 115. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 251.T + 2.97e4T^{2} \)
37 \( 1 + (-37.8 + 65.6i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (83.1 - 144. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-15.5 - 26.9i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 136.T + 1.03e5T^{2} \)
53 \( 1 - 493.T + 1.48e5T^{2} \)
59 \( 1 + (48.0 + 83.2i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (313. + 542. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (496. - 859. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-71.0 - 122. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + 997.T + 3.89e5T^{2} \)
79 \( 1 - 1.20e3T + 4.93e5T^{2} \)
83 \( 1 - 1.12e3T + 5.71e5T^{2} \)
89 \( 1 + (426. - 738. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-357. - 619. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−12.40616244976289493013004907053, −11.69249198690370100733890225119, −11.23937785735164337370427684236, −10.20874141566709044082993160506, −9.193223386769946810308780345994, −7.63463180006916554736480030798, −5.46024782925361349080477297114, −4.59084559558799514577929538075, −3.78405929718837266816388776972, −2.17377077988316592736118397537, 0.18479692945301112013378798543, 3.71893317871230818721724520528, 4.64365813847717486174507454975, 6.00441294685620934633326826343, 7.21363243399301041121856490429, 7.57230634048031233661216213408, 8.528578796256427585868594873436, 10.64364862215268574507970750987, 12.18980669285992597594810416004, 12.37316595921954241492112839434

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