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Properties

Label 2-143-13.9-c3-0-12
Degree $2$
Conductor $143$
Sign $-0.970 + 0.241i$
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  + (−2.09 + 3.62i)2-s + (−2.35 + 4.07i)3-s + (−4.78 − 8.28i)4-s + 14.4·5-s + (−9.85 − 17.0i)6-s + (17.3 + 30.0i)7-s + 6.55·8-s + (2.42 + 4.20i)9-s + (−30.2 + 52.3i)10-s + (−5.5 + 9.52i)11-s + 44.9·12-s + (40.3 + 23.9i)13-s − 145.·14-s + (−33.9 + 58.7i)15-s + (24.5 − 42.4i)16-s + (37.6 + 65.1i)17-s + ⋯
L(s)  = 1  + (−0.740 + 1.28i)2-s + (−0.452 + 0.784i)3-s + (−0.597 − 1.03i)4-s + 1.29·5-s + (−0.670 − 1.16i)6-s + (0.936 + 1.62i)7-s + 0.289·8-s + (0.0899 + 0.155i)9-s + (−0.955 + 1.65i)10-s + (−0.150 + 0.261i)11-s + 1.08·12-s + (0.859 + 0.510i)13-s − 2.77·14-s + (−0.584 + 1.01i)15-s + (0.383 − 0.663i)16-s + (0.536 + 0.929i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 + 0.241i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.970 + 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $-0.970 + 0.241i$
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: \((2,\ 143,\ (\ :3/2),\ -0.970 + 0.241i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.158073 - 1.29229i\)
\(L(\frac12)\) \(\approx\) \(0.158073 - 1.29229i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-40.3 - 23.9i)T \)
good2 \( 1 + (2.09 - 3.62i)T + (-4 - 6.92i)T^{2} \)
3 \( 1 + (2.35 - 4.07i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 - 14.4T + 125T^{2} \)
7 \( 1 + (-17.3 - 30.0i)T + (-171.5 + 297. i)T^{2} \)
17 \( 1 + (-37.6 - 65.1i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (25.2 + 43.7i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-85.1 + 147. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-132. + 230. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 134.T + 2.97e4T^{2} \)
37 \( 1 + (-3.68 + 6.37i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (53.3 - 92.4i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (218. + 378. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 374.T + 1.03e5T^{2} \)
53 \( 1 - 650.T + 1.48e5T^{2} \)
59 \( 1 + (139. + 241. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (26.2 + 45.5i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-42.5 + 73.6i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (132. + 229. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 269.T + 3.89e5T^{2} \)
79 \( 1 - 214.T + 4.93e5T^{2} \)
83 \( 1 - 1.12e3T + 5.71e5T^{2} \)
89 \( 1 + (-436. + 756. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-260. - 450. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−13.45582465682039201185992255828, −12.06702334845405330234796639729, −10.80652223531955581432327615740, −9.805825193559471206556667454619, −8.896613532211536321080866629731, −8.212762345359378623695669936745, −6.45121675705809399679422501826, −5.69527861637483772340928933303, −4.89330898328725831553619620228, −2.07090242257172213116582337859, 1.00800062617897827644612254941, 1.52482044546738038747975230353, 3.45416016331186751252937283128, 5.42151422417050048265719531198, 6.80125790337097734662027369116, 7.978597424551911952729950178976, 9.341857603567138527396330276394, 10.30020223996668913972953690438, 10.93662875058424465446709497193, 11.83125819447516988216827183825

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