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Properties

Label 1-143-143.69-r0-0-0
Degree $1$
Conductor $143$
Sign $0.947 - 0.319i$
Analytic cond. $0.664089$
Root an. cond. $0.664089$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.978 − 0.207i)2-s + (0.913 + 0.406i)3-s + (0.913 − 0.406i)4-s + (−0.309 − 0.951i)5-s + (0.978 + 0.207i)6-s + (−0.913 + 0.406i)7-s + (0.809 − 0.587i)8-s + (0.669 + 0.743i)9-s + (−0.5 − 0.866i)10-s + 12-s + (−0.809 + 0.587i)14-s + (0.104 − 0.994i)15-s + (0.669 − 0.743i)16-s + (−0.978 − 0.207i)17-s + (0.809 + 0.587i)18-s + (0.104 + 0.994i)19-s + ⋯
L(s)  = 1  + (0.978 − 0.207i)2-s + (0.913 + 0.406i)3-s + (0.913 − 0.406i)4-s + (−0.309 − 0.951i)5-s + (0.978 + 0.207i)6-s + (−0.913 + 0.406i)7-s + (0.809 − 0.587i)8-s + (0.669 + 0.743i)9-s + (−0.5 − 0.866i)10-s + 12-s + (−0.809 + 0.587i)14-s + (0.104 − 0.994i)15-s + (0.669 − 0.743i)16-s + (−0.978 − 0.207i)17-s + (0.809 + 0.587i)18-s + (0.104 + 0.994i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.947 - 0.319i$
Analytic conductor: \(0.664089\)
Root analytic conductor: \(0.664089\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 143,\ (0:\ ),\ 0.947 - 0.319i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.207159419 - 0.3619782001i\)
\(L(\frac12)\) \(\approx\) \(2.207159419 - 0.3619782001i\)
\(L(1)\) \(\approx\) \(2.007208961 - 0.2265753474i\)
\(L(1)\) \(\approx\) \(2.007208961 - 0.2265753474i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.978 - 0.207i)T \)
3 \( 1 + (0.913 + 0.406i)T \)
5 \( 1 + (-0.309 - 0.951i)T \)
7 \( 1 + (-0.913 + 0.406i)T \)
17 \( 1 + (-0.978 - 0.207i)T \)
19 \( 1 + (0.104 + 0.994i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.104 + 0.994i)T \)
31 \( 1 + (-0.309 + 0.951i)T \)
37 \( 1 + (0.104 - 0.994i)T \)
41 \( 1 + (-0.913 - 0.406i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (0.809 - 0.587i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (-0.913 + 0.406i)T \)
61 \( 1 + (-0.978 - 0.207i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (0.978 + 0.207i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (0.309 - 0.951i)T \)
83 \( 1 + (-0.309 - 0.951i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.669 - 0.743i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−28.78309916779488745256699629158, −26.86263072894034112132988598075, −26.050391747575399433458315365141, −25.5393829108595995472851654033, −24.24716413194806004415405284617, −23.4841357785978529714777975585, −22.4099376346339561849837418543, −21.656717276712980978800475128754, −20.18434564290344498532392032761, −19.671696810008149865666685516609, −18.58774617870615223220437297868, −17.14681644679538080385988054155, −15.46897394618328334420830568369, −15.29714554492422713482462291509, −13.74508044522191058803199047138, −13.45708633921279384893429811179, −12.12351004051590417353971641733, −10.92206350646387626471608903314, −9.58350848881844364782284502297, −7.92993852933924262289760547210, −6.99114284475981500358975988086, −6.25346026377138227564599538081, −4.1919809460619257100640983206, −3.27023058379995419619918574897, −2.27926305337656735693354654943, 1.93599533071452815945308069933, 3.29187768325526042916419335885, 4.26510053422484713721309178191, 5.423282158255896352572431767586, 6.911052620878134499892856734022, 8.37412183207017433904174356668, 9.44073334262173551794391859784, 10.60349639409105416342935931476, 12.189150141682632093881139831334, 12.88298772701191077354448369293, 13.85757131313696475994542059748, 14.98505015349191914309360433383, 15.98868158264168506493251583921, 16.445399620164494386915820658, 18.64964273265834965350584393721, 19.83635931709759532264182954524, 20.18711170136217008807919454726, 21.30621151496892393914719786472, 22.14835156792100106244565623752, 23.22510135307562174178567335253, 24.49928127235263155569853634756, 25.01026963733890584537767226447, 26.04374595696539213092504581725, 27.33542073493201847632793612475, 28.47035902695945628201920990975

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