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Properties

Label 2-143-13.10-c3-0-19
Degree $2$
Conductor $143$
Sign $0.870 + 0.492i$
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  + (−1.40 + 0.809i)2-s + (1.16 + 2.01i)3-s + (−2.69 + 4.66i)4-s − 8.11i·5-s + (−3.26 − 1.88i)6-s + (−7.84 − 4.53i)7-s − 21.6i·8-s + (10.7 − 18.6i)9-s + (6.56 + 11.3i)10-s + (−9.52 + 5.5i)11-s − 12.5·12-s + (40.0 − 24.3i)13-s + 14.6·14-s + (16.3 − 9.44i)15-s + (−4.01 − 6.94i)16-s + (−1.63 + 2.82i)17-s + ⋯
L(s)  = 1  + (−0.495 + 0.286i)2-s + (0.223 + 0.387i)3-s + (−0.336 + 0.582i)4-s − 0.726i·5-s + (−0.221 − 0.128i)6-s + (−0.423 − 0.244i)7-s − 0.956i·8-s + (0.399 − 0.692i)9-s + (0.207 + 0.359i)10-s + (−0.261 + 0.150i)11-s − 0.301·12-s + (0.854 − 0.518i)13-s + 0.279·14-s + (0.281 − 0.162i)15-s + (−0.0626 − 0.108i)16-s + (−0.0233 + 0.0403i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.870 + 0.492i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.870 + 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.870 + 0.492i$
Analytic conductor: \(8.43727\)
Root analytic conductor: \(2.90469\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 143,\ (\ :3/2),\ 0.870 + 0.492i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.06228 - 0.279503i\)
\(L(\frac12)\) \(\approx\) \(1.06228 - 0.279503i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (9.52 - 5.5i)T \)
13 \( 1 + (-40.0 + 24.3i)T \)
good2 \( 1 + (1.40 - 0.809i)T + (4 - 6.92i)T^{2} \)
3 \( 1 + (-1.16 - 2.01i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + 8.11iT - 125T^{2} \)
7 \( 1 + (7.84 + 4.53i)T + (171.5 + 297. i)T^{2} \)
17 \( 1 + (1.63 - 2.82i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-81.2 - 46.9i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (23.9 + 41.4i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (115. + 199. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 149. iT - 2.97e4T^{2} \)
37 \( 1 + (-308. + 178. i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-105. + 61.0i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (181. - 314. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 292. iT - 1.03e5T^{2} \)
53 \( 1 + 713.T + 1.48e5T^{2} \)
59 \( 1 + (-317. - 183. i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-146. + 253. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (556. - 321. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-93.5 - 54.0i)T + (1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 619. iT - 3.89e5T^{2} \)
79 \( 1 + 1.18e3T + 4.93e5T^{2} \)
83 \( 1 + 1.08e3iT - 5.71e5T^{2} \)
89 \( 1 + (-776. + 448. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-489. - 282. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−12.87341547485856849886755063322, −11.63667724207938698211264287568, −10.03086800796579890810373437115, −9.426616661326051425299153454574, −8.428854584455210625488639832076, −7.50861717876314334952629883402, −6.08806895138428389105998528325, −4.39271608688255825070166219634, −3.39983377438183999564811063569, −0.70271409657757799157757891032, 1.45790501041242875101970323377, 2.99402657453760442056009012963, 4.95099415632350785456217430890, 6.30086122660991939780038586808, 7.47536151522600351146748973145, 8.705125499894752096439306002959, 9.655634154614989523833402712742, 10.70239479141846738564155863668, 11.34734075977643314528778840279, 12.88221212569288871508621078702

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