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Properties

Label 2-143-13.12-c3-0-24
Degree $2$
Conductor $143$
Sign $0.444 + 0.895i$
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  − 1.26i·2-s + 3.96·3-s + 6.38·4-s + 4.99i·5-s − 5.03i·6-s − 32.3i·7-s − 18.2i·8-s − 11.2·9-s + 6.34·10-s + 11i·11-s + 25.3·12-s + (41.9 − 20.8i)13-s − 41.1·14-s + 19.8i·15-s + 27.8·16-s + 65.4·17-s + ⋯
L(s)  = 1  − 0.448i·2-s + 0.763·3-s + 0.798·4-s + 0.447i·5-s − 0.342i·6-s − 1.74i·7-s − 0.807i·8-s − 0.416·9-s + 0.200·10-s + 0.301i·11-s + 0.609·12-s + (0.895 − 0.444i)13-s − 0.784·14-s + 0.341i·15-s + 0.435·16-s + 0.933·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.444 + 0.895i$
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: \((2,\ 143,\ (\ :3/2),\ 0.444 + 0.895i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.12763 - 1.31878i\)
\(L(\frac12)\) \(\approx\) \(2.12763 - 1.31878i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-41.9 + 20.8i)T \)
good2 \( 1 + 1.26iT - 8T^{2} \)
3 \( 1 - 3.96T + 27T^{2} \)
5 \( 1 - 4.99iT - 125T^{2} \)
7 \( 1 + 32.3iT - 343T^{2} \)
17 \( 1 - 65.4T + 4.91e3T^{2} \)
19 \( 1 - 66.3iT - 6.85e3T^{2} \)
23 \( 1 + 25.5T + 1.21e4T^{2} \)
29 \( 1 + 195.T + 2.43e4T^{2} \)
31 \( 1 - 23.5iT - 2.97e4T^{2} \)
37 \( 1 - 95.7iT - 5.06e4T^{2} \)
41 \( 1 - 245. iT - 6.89e4T^{2} \)
43 \( 1 - 397.T + 7.95e4T^{2} \)
47 \( 1 - 139. iT - 1.03e5T^{2} \)
53 \( 1 + 224.T + 1.48e5T^{2} \)
59 \( 1 - 584. iT - 2.05e5T^{2} \)
61 \( 1 + 732.T + 2.26e5T^{2} \)
67 \( 1 + 211. iT - 3.00e5T^{2} \)
71 \( 1 + 59.0iT - 3.57e5T^{2} \)
73 \( 1 + 19.3iT - 3.89e5T^{2} \)
79 \( 1 - 282.T + 4.93e5T^{2} \)
83 \( 1 - 1.35e3iT - 5.71e5T^{2} \)
89 \( 1 - 856. iT - 7.04e5T^{2} \)
97 \( 1 + 1.01e3iT - 9.12e5T^{2} \)
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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

−12.49658648486617416792198649223, −11.13736263971603182547942657572, −10.59708240781889459580965033022, −9.649337173475422620446975764550, −7.979725820064876248119178187425, −7.32786344877823481377152743193, −6.07191685200996731547491608233, −3.86520929262436797590317344353, −3.05365086061245470089880467568, −1.29329365846578273886041285157, 2.05538142901035882383407880355, 3.18723014304047680961147860925, 5.40841240721505213692426249031, 6.13771344353687440140844314830, 7.69555391801122274254233398765, 8.712326938504928903619058774742, 9.169693111973499698868278511773, 11.02833500859091772735341332660, 11.81973354646959553855144326021, 12.76766623880581166547885145259

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