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Properties

Label 2-143-11.3-c1-0-3
Degree $2$
Conductor $143$
Sign $-0.00539 - 0.999i$
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  + (0.203 − 0.148i)2-s + (0.947 + 2.91i)3-s + (−0.598 + 1.84i)4-s + (−1.49 − 1.08i)5-s + (0.625 + 0.454i)6-s + (0.992 − 3.05i)7-s + (0.306 + 0.943i)8-s + (−5.17 + 3.75i)9-s − 0.464·10-s + (3.18 + 0.932i)11-s − 5.93·12-s + (−0.809 + 0.587i)13-s + (−0.250 − 0.769i)14-s + (1.74 − 5.37i)15-s + (−2.93 − 2.12i)16-s + (4.91 + 3.57i)17-s + ⋯
L(s)  = 1  + (0.144 − 0.104i)2-s + (0.546 + 1.68i)3-s + (−0.299 + 0.920i)4-s + (−0.667 − 0.484i)5-s + (0.255 + 0.185i)6-s + (0.375 − 1.15i)7-s + (0.108 + 0.333i)8-s + (−1.72 + 1.25i)9-s − 0.147·10-s + (0.959 + 0.281i)11-s − 1.71·12-s + (−0.224 + 0.163i)13-s + (−0.0668 − 0.205i)14-s + (0.451 − 1.38i)15-s + (−0.732 − 0.532i)16-s + (1.19 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00539 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00539 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $-0.00539 - 0.999i$
Analytic conductor: \(1.14186\)
Root analytic conductor: \(1.06857\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 143,\ (\ :1/2),\ -0.00539 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.865484 + 0.870169i\)
\(L(\frac12)\) \(\approx\) \(0.865484 + 0.870169i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-3.18 - 0.932i)T \)
13 \( 1 + (0.809 - 0.587i)T \)
good2 \( 1 + (-0.203 + 0.148i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (-0.947 - 2.91i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (1.49 + 1.08i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.992 + 3.05i)T + (-5.66 - 4.11i)T^{2} \)
17 \( 1 + (-4.91 - 3.57i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.04 + 3.23i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 7.01T + 23T^{2} \)
29 \( 1 + (-1.18 + 3.65i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (3.96 - 2.87i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.477 + 1.46i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.69 + 5.22i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 6.07T + 43T^{2} \)
47 \( 1 + (1.32 + 4.06i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (1.82 - 1.32i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.741 - 2.28i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-1.14 - 0.830i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 15.7T + 67T^{2} \)
71 \( 1 + (-3.62 - 2.63i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.860 + 2.64i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-7.17 + 5.21i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (7.23 + 5.25i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 0.928T + 89T^{2} \)
97 \( 1 + (8.68 - 6.30i)T + (29.9 - 92.2i)T^{2} \)
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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

−13.54065729850182039092030604029, −12.25507120872740477074542722588, −11.25047273493332078534853992407, −10.28526381149932628918738828527, −9.142708066529211351981521390466, −8.398108518412395890268602278223, −7.32994869998865437247522529184, −4.86034933962234191445870893134, −4.15310475624554539652507898970, −3.41225158150521799014924601000, 1.43647888747638155477378831151, 3.11828501873792434540733583579, 5.38181029613448471241147172028, 6.46943141474335757297359650360, 7.43992898420710224937359295712, 8.531359882086489373712636611903, 9.456267644116841890667742964319, 11.25380723864231643583017667719, 11.97148579937840986798415285867, 12.87484129637263568432288665569

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