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Properties

Label 2-143-13.12-c3-0-1
Degree $2$
Conductor $143$
Sign $-0.529 + 0.848i$
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  + 3.97i·2-s − 1.00·3-s − 7.81·4-s − 0.345i·5-s − 3.98i·6-s + 17.7i·7-s + 0.755i·8-s − 25.9·9-s + 1.37·10-s − 11i·11-s + 7.83·12-s + (−39.7 − 24.7i)13-s − 70.5·14-s + 0.346i·15-s − 65.4·16-s + 18.9·17-s + ⋯
L(s)  = 1  + 1.40i·2-s − 0.193·3-s − 0.976·4-s − 0.0308i·5-s − 0.271i·6-s + 0.958i·7-s + 0.0333i·8-s − 0.962·9-s + 0.0434·10-s − 0.301i·11-s + 0.188·12-s + (−0.848 − 0.529i)13-s − 1.34·14-s + 0.00596i·15-s − 1.02·16-s + 0.270·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.529 + 0.848i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.529 + 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $-0.529 + 0.848i$
Analytic conductor: \(8.43727\)
Root analytic conductor: \(2.90469\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 143,\ (\ :3/2),\ -0.529 + 0.848i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.318812 - 0.574474i\)
\(L(\frac12)\) \(\approx\) \(0.318812 - 0.574474i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + 11iT \)
13 \( 1 + (39.7 + 24.7i)T \)
good2 \( 1 - 3.97iT - 8T^{2} \)
3 \( 1 + 1.00T + 27T^{2} \)
5 \( 1 + 0.345iT - 125T^{2} \)
7 \( 1 - 17.7iT - 343T^{2} \)
17 \( 1 - 18.9T + 4.91e3T^{2} \)
19 \( 1 + 17.8iT - 6.85e3T^{2} \)
23 \( 1 + 196.T + 1.21e4T^{2} \)
29 \( 1 - 94.8T + 2.43e4T^{2} \)
31 \( 1 - 115. iT - 2.97e4T^{2} \)
37 \( 1 - 215. iT - 5.06e4T^{2} \)
41 \( 1 - 338. iT - 6.89e4T^{2} \)
43 \( 1 + 390.T + 7.95e4T^{2} \)
47 \( 1 + 439. iT - 1.03e5T^{2} \)
53 \( 1 + 585.T + 1.48e5T^{2} \)
59 \( 1 - 796. iT - 2.05e5T^{2} \)
61 \( 1 - 347.T + 2.26e5T^{2} \)
67 \( 1 - 868. iT - 3.00e5T^{2} \)
71 \( 1 - 792. iT - 3.57e5T^{2} \)
73 \( 1 + 342. iT - 3.89e5T^{2} \)
79 \( 1 - 868.T + 4.93e5T^{2} \)
83 \( 1 - 1.01e3iT - 5.71e5T^{2} \)
89 \( 1 - 472. iT - 7.04e5T^{2} \)
97 \( 1 + 1.03e3iT - 9.12e5T^{2} \)
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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

−13.67736782187738000751743050990, −12.27303910528006846770889673900, −11.49885205040817603497918219931, −10.01251607579895891232967495431, −8.640117259546233070606917397431, −8.128534296802534607342569945918, −6.71534380218275239408668897227, −5.75931931450658727548006772719, −4.98141172668241255284313783126, −2.73246492167633754108733061003, 0.31082937917256268129119383233, 2.08800124419272009791916284142, 3.52839383427264743769569343933, 4.73813830303420615620628904864, 6.47493419440676062927841451906, 7.81761624698043664448468239877, 9.299167754212244254687763056970, 10.20027511370292350173932366650, 10.98052637673624088544792856787, 11.92901903874071325994819268774

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