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Properties

Label 2-143-11.4-c1-0-9
Degree $2$
Conductor $143$
Sign $-0.553 + 0.832i$
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  + (−1.03 − 0.754i)2-s + (0.598 − 1.84i)3-s + (−0.109 − 0.335i)4-s + (2.14 − 1.55i)5-s + (−2.01 + 1.46i)6-s + (0.230 + 0.708i)7-s + (−0.933 + 2.87i)8-s + (−0.612 − 0.444i)9-s − 3.39·10-s + (−3.17 − 0.969i)11-s − 0.683·12-s + (0.809 + 0.587i)13-s + (0.295 − 0.909i)14-s + (−1.58 − 4.88i)15-s + (2.56 − 1.86i)16-s + (2.09 − 1.52i)17-s + ⋯
L(s)  = 1  + (−0.734 − 0.533i)2-s + (0.345 − 1.06i)3-s + (−0.0545 − 0.167i)4-s + (0.957 − 0.695i)5-s + (−0.821 + 0.596i)6-s + (0.0870 + 0.267i)7-s + (−0.329 + 1.01i)8-s + (−0.204 − 0.148i)9-s − 1.07·10-s + (−0.956 − 0.292i)11-s − 0.197·12-s + (0.224 + 0.163i)13-s + (0.0789 − 0.243i)14-s + (−0.409 − 1.26i)15-s + (0.641 − 0.465i)16-s + (0.507 − 0.368i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.553 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $-0.553 + 0.832i$
Analytic conductor: \(1.14186\)
Root analytic conductor: \(1.06857\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (92, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 143,\ (\ :1/2),\ -0.553 + 0.832i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.433569 - 0.809162i\)
\(L(\frac12)\) \(\approx\) \(0.433569 - 0.809162i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (3.17 + 0.969i)T \)
13 \( 1 + (-0.809 - 0.587i)T \)
good2 \( 1 + (1.03 + 0.754i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (-0.598 + 1.84i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (-2.14 + 1.55i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (-0.230 - 0.708i)T + (-5.66 + 4.11i)T^{2} \)
17 \( 1 + (-2.09 + 1.52i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.34 - 4.14i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 2.54T + 23T^{2} \)
29 \( 1 + (0.593 + 1.82i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-4.92 - 3.57i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.68 - 8.25i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-3.11 + 9.57i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 0.689T + 43T^{2} \)
47 \( 1 + (1.99 - 6.13i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (5.94 + 4.32i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.377 + 1.16i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-1.52 + 1.11i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 11.2T + 67T^{2} \)
71 \( 1 + (-11.2 + 8.14i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-3.90 - 12.0i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (6.64 + 4.82i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (12.4 - 9.01i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 6.22T + 89T^{2} \)
97 \( 1 + (-2.97 - 2.16i)T + (29.9 + 92.2i)T^{2} \)
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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

−12.77352207326203285900293314933, −11.85685673073489488905808771706, −10.47039478019436886145782921680, −9.704121518655648674824047259119, −8.573113488553209134457416944636, −7.88054832360181497480760152636, −6.17319302755383943062546707693, −5.17544665563195935638546005200, −2.44649822960587382305316750625, −1.34178406608711536484956939095, 2.89651174103091240593398761500, 4.35120558849697675800090673385, 5.99452679656907369343515555978, 7.23651388867622023232491685650, 8.352035811823305781689187787229, 9.508135899334424174616401534458, 10.05999285255476021036066391304, 10.87190230213657513879891187603, 12.65522968625443388129552088051, 13.59066991135648798097610380612

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