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Properties

Label 1-429-429.20-r0-0-0
Degree $1$
Conductor $429$
Sign $0.436 + 0.899i$
Analytic cond. $1.99226$
Root an. cond. $1.99226$
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  + (0.994 + 0.104i)2-s + (0.978 + 0.207i)4-s + (0.587 + 0.809i)5-s + (−0.207 + 0.978i)7-s + (0.951 + 0.309i)8-s + (0.5 + 0.866i)10-s + (−0.309 + 0.951i)14-s + (0.913 + 0.406i)16-s + (−0.104 − 0.994i)17-s + (−0.743 + 0.669i)19-s + (0.406 + 0.913i)20-s + (−0.5 − 0.866i)23-s + (−0.309 + 0.951i)25-s + (−0.406 + 0.913i)28-s + (−0.669 + 0.743i)29-s + ⋯
L(s)  = 1  + (0.994 + 0.104i)2-s + (0.978 + 0.207i)4-s + (0.587 + 0.809i)5-s + (−0.207 + 0.978i)7-s + (0.951 + 0.309i)8-s + (0.5 + 0.866i)10-s + (−0.309 + 0.951i)14-s + (0.913 + 0.406i)16-s + (−0.104 − 0.994i)17-s + (−0.743 + 0.669i)19-s + (0.406 + 0.913i)20-s + (−0.5 − 0.866i)23-s + (−0.309 + 0.951i)25-s + (−0.406 + 0.913i)28-s + (−0.669 + 0.743i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.436 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.436 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.436 + 0.899i$
Analytic conductor: \(1.99226\)
Root analytic conductor: \(1.99226\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 429,\ (0:\ ),\ 0.436 + 0.899i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.266170942 + 1.418798173i\)
\(L(\frac12)\) \(\approx\) \(2.266170942 + 1.418798173i\)
\(L(1)\) \(\approx\) \(1.913489720 + 0.6403680751i\)
\(L(1)\) \(\approx\) \(1.913489720 + 0.6403680751i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.994 + 0.104i)T \)
5 \( 1 + (0.587 + 0.809i)T \)
7 \( 1 + (-0.207 + 0.978i)T \)
17 \( 1 + (-0.104 - 0.994i)T \)
19 \( 1 + (-0.743 + 0.669i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.669 + 0.743i)T \)
31 \( 1 + (0.587 - 0.809i)T \)
37 \( 1 + (-0.743 - 0.669i)T \)
41 \( 1 + (0.207 + 0.978i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (0.951 + 0.309i)T \)
53 \( 1 + (0.809 + 0.587i)T \)
59 \( 1 + (0.207 - 0.978i)T \)
61 \( 1 + (-0.104 - 0.994i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + (-0.994 + 0.104i)T \)
73 \( 1 + (0.951 - 0.309i)T \)
79 \( 1 + (-0.809 - 0.587i)T \)
83 \( 1 + (0.587 + 0.809i)T \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 + (0.406 + 0.913i)T \)
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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

−23.950667027293280703861703082380, −23.24049816228590049205088482919, −22.242241443278130344982560399885, −21.360758475770703487121409799414, −20.780486039306050346320190347053, −19.83110394397349474594158052462, −19.297646616906200139418667859101, −17.526223269163562265697193673177, −17.01335353950412769015002101466, −16.069665481843004092551399477253, −15.18403194478279799375306743410, −14.0483534810900005625729915397, −13.39308422799793721474900275987, −12.782217552192898088649594046631, −11.77684282726986102930920807344, −10.63324481301977138746461892843, −9.9624925532022026788240312811, −8.64379686255434711103971200993, −7.44886341485956976380993208845, −6.42240256932006336728519713841, −5.55166646391162537079763471942, −4.46267968089174041880074542638, −3.748275915149841095690084286666, −2.28235633115213381507964415853, −1.190087425085489707446420190048, 2.03296148463048624825476204745, 2.672108380701867242680841997360, 3.79104685595335694705371174375, 5.11407400443692606656803654444, 5.98596110598670239140754252567, 6.66700227664730826417361898379, 7.76623623701522391541677582509, 9.08248352319818506042606367638, 10.22184997872013627804282065790, 11.111132482642477837307082937380, 12.075576813822529772171366795177, 12.85699821271178274262154156382, 13.90294484725260154969421671345, 14.574680705098618085246893039, 15.35861768536179166640835593467, 16.21986060953388324445287387308, 17.25196984579505118384537161860, 18.42018402624358663316298453793, 19.006917475870797382312325627309, 20.302387463018621868420814526481, 21.10771451569067449025600673689, 21.92219556047991950030563532650, 22.487284484823619221582345851755, 23.186818539562685160334837773370, 24.423407355484355251996054367088

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