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Properties

Label 2-143-13.12-c3-0-25
Degree $2$
Conductor $143$
Sign $0.876 - 0.480i$
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  + 3.37i·2-s + 7.92·3-s − 3.41·4-s − 15.9i·5-s + 26.7i·6-s − 29.7i·7-s + 15.4i·8-s + 35.8·9-s + 53.9·10-s − 11i·11-s − 27.0·12-s + (22.5 + 41.0i)13-s + 100.·14-s − 126. i·15-s − 79.6·16-s + 67.2·17-s + ⋯
L(s)  = 1  + 1.19i·2-s + 1.52·3-s − 0.427·4-s − 1.42i·5-s + 1.82i·6-s − 1.60i·7-s + 0.684i·8-s + 1.32·9-s + 1.70·10-s − 0.301i·11-s − 0.651·12-s + (0.480 + 0.876i)13-s + 1.91·14-s − 2.17i·15-s − 1.24·16-s + 0.958·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 - 0.480i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.876 - 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.876 - 0.480i$
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: \((2,\ 143,\ (\ :3/2),\ 0.876 - 0.480i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.74616 + 0.703589i\)
\(L(\frac12)\) \(\approx\) \(2.74616 + 0.703589i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-22.5 - 41.0i)T \)
good2 \( 1 - 3.37iT - 8T^{2} \)
3 \( 1 - 7.92T + 27T^{2} \)
5 \( 1 + 15.9iT - 125T^{2} \)
7 \( 1 + 29.7iT - 343T^{2} \)
17 \( 1 - 67.2T + 4.91e3T^{2} \)
19 \( 1 - 45.1iT - 6.85e3T^{2} \)
23 \( 1 + 162.T + 1.21e4T^{2} \)
29 \( 1 - 155.T + 2.43e4T^{2} \)
31 \( 1 - 235. iT - 2.97e4T^{2} \)
37 \( 1 - 64.5iT - 5.06e4T^{2} \)
41 \( 1 - 303. iT - 6.89e4T^{2} \)
43 \( 1 + 154.T + 7.95e4T^{2} \)
47 \( 1 + 18.5iT - 1.03e5T^{2} \)
53 \( 1 - 289.T + 1.48e5T^{2} \)
59 \( 1 + 345. iT - 2.05e5T^{2} \)
61 \( 1 + 428.T + 2.26e5T^{2} \)
67 \( 1 - 114. iT - 3.00e5T^{2} \)
71 \( 1 + 976. iT - 3.57e5T^{2} \)
73 \( 1 + 206. iT - 3.89e5T^{2} \)
79 \( 1 - 998.T + 4.93e5T^{2} \)
83 \( 1 - 895. iT - 5.71e5T^{2} \)
89 \( 1 + 611. iT - 7.04e5T^{2} \)
97 \( 1 - 1.17e3iT - 9.12e5T^{2} \)
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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

−13.34087190811017619881501434565, −11.98961799518759565874657741268, −10.26773833159335710962222641208, −9.130439362075471261333641714581, −8.194818711549590469048803943138, −7.78149952418504111891787095921, −6.47087202482342259717634382315, −4.79021985341023772317051163806, −3.70444926991475460023878332176, −1.44719944886434603825620981574, 2.19595622280699701127918750186, 2.77225576543039554553426768151, 3.66276422941667131337934237407, 6.03805785676049092103169844245, 7.50170851455380705758438977815, 8.596106072800598193574806502994, 9.695950503664241398430469474458, 10.38059117645036444278491813301, 11.60299601739748905290444883922, 12.42669060551634764930686779605

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