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Properties

Label 2-143-13.12-c3-0-27
Degree $2$
Conductor $143$
Sign $-0.0256 + 0.999i$
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  − 2.40i·2-s + 7.21·3-s + 2.21·4-s − 9.84i·5-s − 17.3i·6-s + 8.26i·7-s − 24.5i·8-s + 25.1·9-s − 23.6·10-s − 11i·11-s + 16.0·12-s + (−46.8 − 1.20i)13-s + 19.8·14-s − 71.0i·15-s − 41.3·16-s + 113.·17-s + ⋯
L(s)  = 1  − 0.850i·2-s + 1.38·3-s + 0.277·4-s − 0.880i·5-s − 1.18i·6-s + 0.446i·7-s − 1.08i·8-s + 0.929·9-s − 0.748·10-s − 0.301i·11-s + 0.385·12-s + (−0.999 − 0.0256i)13-s + 0.379·14-s − 1.22i·15-s − 0.645·16-s + 1.61·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0256 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0256 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $-0.0256 + 0.999i$
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: \((2,\ 143,\ (\ :3/2),\ -0.0256 + 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.98648 - 2.03817i\)
\(L(\frac12)\) \(\approx\) \(1.98648 - 2.03817i\)
\(L(\frac{5}{2})\) 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 \( 1 + (46.8 + 1.20i)T \)
good2 \( 1 + 2.40iT - 8T^{2} \)
3 \( 1 - 7.21T + 27T^{2} \)
5 \( 1 + 9.84iT - 125T^{2} \)
7 \( 1 - 8.26iT - 343T^{2} \)
17 \( 1 - 113.T + 4.91e3T^{2} \)
19 \( 1 - 121. iT - 6.85e3T^{2} \)
23 \( 1 - 69.3T + 1.21e4T^{2} \)
29 \( 1 + 209.T + 2.43e4T^{2} \)
31 \( 1 + 18.1iT - 2.97e4T^{2} \)
37 \( 1 + 228. iT - 5.06e4T^{2} \)
41 \( 1 - 460. iT - 6.89e4T^{2} \)
43 \( 1 + 386.T + 7.95e4T^{2} \)
47 \( 1 - 322. iT - 1.03e5T^{2} \)
53 \( 1 - 249.T + 1.48e5T^{2} \)
59 \( 1 + 56.7iT - 2.05e5T^{2} \)
61 \( 1 - 775.T + 2.26e5T^{2} \)
67 \( 1 - 387. iT - 3.00e5T^{2} \)
71 \( 1 + 116. iT - 3.57e5T^{2} \)
73 \( 1 - 1.01e3iT - 3.89e5T^{2} \)
79 \( 1 + 429.T + 4.93e5T^{2} \)
83 \( 1 + 950. iT - 5.71e5T^{2} \)
89 \( 1 - 671. iT - 7.04e5T^{2} \)
97 \( 1 + 968. iT - 9.12e5T^{2} \)
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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

−12.50074024324653957880798449476, −11.57918039964356020436370401302, −10.05398930412937628003515757989, −9.435839003737189363624237359823, −8.324604302837363662365190411924, −7.42136073552410109594142834800, −5.54285344198718795483571571635, −3.78455204629942592860527145212, −2.73956233179933430304771692585, −1.43278476656689998949682312684, 2.32500262556900114279605331382, 3.35049405338650575601182976995, 5.24390109796209517835499586366, 7.06462181060897306448796612201, 7.28410553630005599400989275629, 8.445884531815759979365665892453, 9.626234499006458422950083823812, 10.68529975532639561508560630540, 11.90966844672077056256699368210, 13.38338846074610378026076870177

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