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Properties

Label 2-143-11.5-c1-0-5
Degree $2$
Conductor $143$
Sign $0.949 + 0.312i$
Analytic cond. $1.14186$
Root an. cond. $1.06857$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.248 − 0.765i)2-s + (1.92 + 1.39i)3-s + (1.09 − 0.794i)4-s + (0.172 − 0.531i)5-s + (0.590 − 1.81i)6-s + (−1.38 + 1.00i)7-s + (−2.18 − 1.58i)8-s + (0.814 + 2.50i)9-s − 0.450·10-s + (−2.59 + 2.06i)11-s + 3.20·12-s + (0.309 + 0.951i)13-s + (1.11 + 0.810i)14-s + (1.07 − 0.780i)15-s + (0.163 − 0.503i)16-s + (1.05 − 3.26i)17-s + ⋯
L(s)  = 1  + (−0.175 − 0.541i)2-s + (1.10 + 0.805i)3-s + (0.546 − 0.397i)4-s + (0.0772 − 0.237i)5-s + (0.241 − 0.742i)6-s + (−0.523 + 0.380i)7-s + (−0.771 − 0.560i)8-s + (0.271 + 0.835i)9-s − 0.142·10-s + (−0.783 + 0.621i)11-s + 0.926·12-s + (0.0857 + 0.263i)13-s + (0.297 + 0.216i)14-s + (0.277 − 0.201i)15-s + (0.0409 − 0.125i)16-s + (0.257 − 0.791i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.312i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.949 + 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.949 + 0.312i$
Analytic conductor: \(1.14186\)
Root analytic conductor: \(1.06857\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 143,\ (\ :1/2),\ 0.949 + 0.312i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41232 - 0.226259i\)
\(L(\frac12)\) \(\approx\) \(1.41232 - 0.226259i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (2.59 - 2.06i)T \)
13 \( 1 + (-0.309 - 0.951i)T \)
good2 \( 1 + (0.248 + 0.765i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-1.92 - 1.39i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (-0.172 + 0.531i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (1.38 - 1.00i)T + (2.16 - 6.65i)T^{2} \)
17 \( 1 + (-1.05 + 3.26i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (0.955 + 0.694i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 5.43T + 23T^{2} \)
29 \( 1 + (1.11 - 0.808i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-1.35 - 4.15i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-2.62 + 1.90i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-5.74 - 4.17i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 7.59T + 43T^{2} \)
47 \( 1 + (8.91 + 6.47i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-0.385 - 1.18i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-0.916 + 0.665i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-2.15 + 6.64i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 14.2T + 67T^{2} \)
71 \( 1 + (-2.77 + 8.53i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-9.86 + 7.16i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-2.44 - 7.53i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (0.479 - 1.47i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 0.645T + 89T^{2} \)
97 \( 1 + (-0.0284 - 0.0877i)T + (-78.4 + 57.0i)T^{2} \)
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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

−12.99521754393491041138963751813, −12.02045281895982711468212375177, −10.78742293741720444629932687345, −9.758552372900388062868747617613, −9.371981099486776998750354400933, −8.086937431657849001005184420038, −6.61641285289947667941027988709, −5.04701057592041625176361819127, −3.38887998715201421254591153026, −2.31521109009708949252700049505, 2.38085516360547423221317403543, 3.48071654416521783030464089063, 5.96372805385761776416467818222, 6.93199656887032567635986056574, 8.033219788425088102345875566983, 8.376026804554967478861915285341, 9.956455160884127423976086592986, 11.15111970499642382625001768014, 12.53682716744922640278083223724, 13.18416321952878986258762018514

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