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Properties

Label 2-143-11.4-c1-0-10
Degree $2$
Conductor $143$
Sign $-0.879 - 0.475i$
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  + (−1.91 − 1.38i)2-s + (0.850 − 2.61i)3-s + (1.10 + 3.40i)4-s + (−1.81 + 1.31i)5-s + (−5.26 + 3.82i)6-s + (−1.38 − 4.27i)7-s + (1.15 − 3.54i)8-s + (−3.70 − 2.69i)9-s + 5.29·10-s + (−0.676 + 3.24i)11-s + 9.85·12-s + (−0.809 − 0.587i)13-s + (−3.27 + 10.0i)14-s + (1.90 + 5.86i)15-s + (−1.33 + 0.969i)16-s + (0.419 − 0.304i)17-s + ⋯
L(s)  = 1  + (−1.35 − 0.981i)2-s + (0.491 − 1.51i)3-s + (0.552 + 1.70i)4-s + (−0.810 + 0.589i)5-s + (−2.14 + 1.56i)6-s + (−0.524 − 1.61i)7-s + (0.407 − 1.25i)8-s + (−1.23 − 0.896i)9-s + 1.67·10-s + (−0.204 + 0.978i)11-s + 2.84·12-s + (−0.224 − 0.163i)13-s + (−0.875 + 2.69i)14-s + (0.492 + 1.51i)15-s + (−0.333 + 0.242i)16-s + (0.101 − 0.0738i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.879 - 0.475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.879 - 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $-0.879 - 0.475i$
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: \((2,\ 143,\ (\ :1/2),\ -0.879 - 0.475i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.106714 + 0.421533i\)
\(L(\frac12)\) \(\approx\) \(0.106714 + 0.421533i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (0.676 - 3.24i)T \)
13 \( 1 + (0.809 + 0.587i)T \)
good2 \( 1 + (1.91 + 1.38i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (-0.850 + 2.61i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (1.81 - 1.31i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (1.38 + 4.27i)T + (-5.66 + 4.11i)T^{2} \)
17 \( 1 + (-0.419 + 0.304i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.949 + 2.92i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 1.35T + 23T^{2} \)
29 \( 1 + (2.80 + 8.62i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-3.14 - 2.28i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.43 + 7.48i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.29 + 3.99i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 1.55T + 43T^{2} \)
47 \( 1 + (-2.67 + 8.24i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (2.09 + 1.52i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1.47 - 4.53i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-0.382 + 0.278i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 3.47T + 67T^{2} \)
71 \( 1 + (-8.32 + 6.04i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-4.22 - 12.9i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-6.99 - 5.07i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (10.8 - 7.89i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 4.60T + 89T^{2} \)
97 \( 1 + (-6.24 - 4.53i)T + (29.9 + 92.2i)T^{2} \)
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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

−12.37468369628951310981128576536, −11.42682164497102785518893933957, −10.49171401937668022210730436366, −9.528967891300946247387166831907, −8.112941877878762087060842128813, −7.30918221841157260436257155911, −7.03965526489754879142571225104, −3.69691786535373805531035122282, −2.35228997020105688140429067093, −0.63025568280400490288560303560, 3.27755278432318505455080852010, 5.05128469553371245017049264501, 6.12767708085502080928498978063, 7.974973161934717004672730745143, 8.701082959220608132198754221656, 9.205358550911575853458216381035, 10.10595058354544781972479362816, 11.28282916334120114913377212146, 12.50975475327035751166409590370, 14.39405569245540970324433308189

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