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Properties

Label 1-143-143.25-r0-0-0
Degree $1$
Conductor $143$
Sign $0.331 + 0.943i$
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  + (−0.309 + 0.951i)2-s + (−0.809 + 0.587i)3-s + (−0.809 − 0.587i)4-s + (−0.309 − 0.951i)5-s + (−0.309 − 0.951i)6-s + (0.809 + 0.587i)7-s + (0.809 − 0.587i)8-s + (0.309 − 0.951i)9-s + 10-s + 12-s + (−0.809 + 0.587i)14-s + (0.809 + 0.587i)15-s + (0.309 + 0.951i)16-s + (0.309 + 0.951i)17-s + (0.809 + 0.587i)18-s + (0.809 − 0.587i)19-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)2-s + (−0.809 + 0.587i)3-s + (−0.809 − 0.587i)4-s + (−0.309 − 0.951i)5-s + (−0.309 − 0.951i)6-s + (0.809 + 0.587i)7-s + (0.809 − 0.587i)8-s + (0.309 − 0.951i)9-s + 10-s + 12-s + (−0.809 + 0.587i)14-s + (0.809 + 0.587i)15-s + (0.309 + 0.951i)16-s + (0.309 + 0.951i)17-s + (0.809 + 0.587i)18-s + (0.809 − 0.587i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.331 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.331 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.331 + 0.943i$
Analytic conductor: \(0.664089\)
Root analytic conductor: \(0.664089\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 143,\ (0:\ ),\ 0.331 + 0.943i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5698154167 + 0.4037344713i\)
\(L(\frac12)\) \(\approx\) \(0.5698154167 + 0.4037344713i\)
\(L(1)\) \(\approx\) \(0.6427676411 + 0.3253813563i\)
\(L(1)\) \(\approx\) \(0.6427676411 + 0.3253813563i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.309 + 0.951i)T \)
3 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 + (-0.309 - 0.951i)T \)
7 \( 1 + (0.809 + 0.587i)T \)
17 \( 1 + (0.309 + 0.951i)T \)
19 \( 1 + (0.809 - 0.587i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.809 - 0.587i)T \)
31 \( 1 + (-0.309 + 0.951i)T \)
37 \( 1 + (0.809 + 0.587i)T \)
41 \( 1 + (0.809 - 0.587i)T \)
43 \( 1 + T \)
47 \( 1 + (0.809 - 0.587i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (0.809 + 0.587i)T \)
61 \( 1 + (0.309 + 0.951i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.309 - 0.951i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (0.309 - 0.951i)T \)
83 \( 1 + (-0.309 - 0.951i)T \)
89 \( 1 - T \)
97 \( 1 + (-0.309 + 0.951i)T \)
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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

−28.114600248776143366659447294, −27.24774965148339622682720114091, −26.623153078339937784207705027349, −25.19989996575153358665070925256, −23.835393442052900218563798399279, −22.910939379605145693483226413535, −22.32562449745124228996731704380, −21.10971648342580688461475303357, −20.04323432787257162339750362598, −18.82139231592738730095742288757, −18.288609918836479251907448441, −17.39355860924971232860125996685, −16.34659193602986494044520632973, −14.51148304798762792829667378490, −13.61418501803988404028334707107, −12.36631555189273506634884720344, −11.23986133233471211668600827512, −10.95912106231300727027583272664, −9.63524832989628723554368700351, −7.79862209008182826229652986500, −7.2487713873921671799985541104, −5.44404864388054589275363812804, −4.09491453265842737362613230934, −2.57094623460789239203138680876, −1.07073143529686900259349677684, 1.13377079501859907296241865843, 4.1000243457201644899913113598, 5.11618220995977758773736640410, 5.789462167175465275986398570665, 7.37766266848087566686055569667, 8.62508525538567492773861914203, 9.41027946698982536815297072972, 10.80293525538525275785206279303, 11.962372249114988763251694592790, 13.11914511386623082822632389575, 14.72816461992767832098733852549, 15.481683784276865483214503704187, 16.40819959547236579314522436553, 17.25317596256962873993493150659, 18.008092857705671110604650287494, 19.28221249331680857542034526313, 20.71483134828830015399600408386, 21.6725051075993441538072128126, 22.76549875407580270583275868640, 23.8622074226719266600134906071, 24.30207266666207563795740630974, 25.46289456349977735627696582706, 26.787544384493721726313862267182, 27.47758390018890014151837203793, 28.27603186044223160138425983686

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