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Properties

Label 2-143-13.3-c3-0-6
Degree $2$
Conductor $143$
Sign $-0.928 + 0.370i$
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  + (2.05 + 3.56i)2-s + (2.00 + 3.47i)3-s + (−4.45 + 7.71i)4-s − 12.3·5-s + (−8.25 + 14.3i)6-s + (−14.8 + 25.6i)7-s − 3.74·8-s + (5.43 − 9.41i)9-s + (−25.3 − 43.9i)10-s + (−5.5 − 9.52i)11-s − 35.7·12-s + (18.7 − 42.9i)13-s − 121.·14-s + (−24.7 − 42.9i)15-s + (27.9 + 48.4i)16-s + (−50.0 + 86.6i)17-s + ⋯
L(s)  = 1  + (0.726 + 1.25i)2-s + (0.386 + 0.669i)3-s + (−0.556 + 0.964i)4-s − 1.10·5-s + (−0.561 + 0.972i)6-s + (−0.800 + 1.38i)7-s − 0.165·8-s + (0.201 − 0.348i)9-s + (−0.802 − 1.38i)10-s + (−0.150 − 0.261i)11-s − 0.860·12-s + (0.401 − 0.916i)13-s − 2.32·14-s + (−0.426 − 0.738i)15-s + (0.436 + 0.756i)16-s + (−0.713 + 1.23i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.928 + 0.370i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.928 + 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $-0.928 + 0.370i$
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: \((2,\ 143,\ (\ :3/2),\ -0.928 + 0.370i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.344917 - 1.79755i\)
\(L(\frac12)\) \(\approx\) \(0.344917 - 1.79755i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-18.7 + 42.9i)T \)
good2 \( 1 + (-2.05 - 3.56i)T + (-4 + 6.92i)T^{2} \)
3 \( 1 + (-2.00 - 3.47i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + 12.3T + 125T^{2} \)
7 \( 1 + (14.8 - 25.6i)T + (-171.5 - 297. i)T^{2} \)
17 \( 1 + (50.0 - 86.6i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (19.3 - 33.5i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-53.9 - 93.3i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (68.0 + 117. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 7.90T + 2.97e4T^{2} \)
37 \( 1 + (-207. - 359. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-128. - 222. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-9.52 + 16.4i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 167.T + 1.03e5T^{2} \)
53 \( 1 - 96.3T + 1.48e5T^{2} \)
59 \( 1 + (-260. + 451. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-351. + 608. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-334. - 579. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (258. - 448. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + 1.15e3T + 3.89e5T^{2} \)
79 \( 1 - 1.19e3T + 4.93e5T^{2} \)
83 \( 1 - 77.3T + 5.71e5T^{2} \)
89 \( 1 + (272. + 472. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (331. - 574. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−13.19291102752597516879379468523, −12.61378612224891401435840218964, −11.34892904416026260696548934788, −9.929341427517446319588849129742, −8.643438014961297889238591269619, −7.982600312033250613645405185769, −6.49795084068227828883591711465, −5.64899597869302254873311666755, −4.20547906976397241605968918509, −3.27719807551657945924005460550, 0.69825876821397173081980150495, 2.45852190789000869852852764711, 3.84084105473280205991901666723, 4.55665178112122987973691557028, 7.01636885364240351694083451612, 7.46656105970789825756866900167, 9.131346423552619663847084326695, 10.53919723689104566125438867730, 11.15030418692938852299328679178, 12.20438280674061306868745255230

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