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Properties

Label 1-432-432.5-r1-0-0
Degree $1$
Conductor $432$
Sign $0.512 - 0.858i$
Analytic cond. $46.4248$
Root an. cond. $46.4248$
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  + (−0.642 − 0.766i)5-s + (0.939 − 0.342i)7-s + (0.642 − 0.766i)11-s + (0.984 − 0.173i)13-s + (0.5 + 0.866i)17-s + (0.866 + 0.5i)19-s + (−0.939 − 0.342i)23-s + (−0.173 + 0.984i)25-s + (0.984 + 0.173i)29-s + (−0.939 − 0.342i)31-s + (−0.866 − 0.5i)35-s + (0.866 − 0.5i)37-s + (0.173 + 0.984i)41-s + (−0.642 + 0.766i)43-s + (0.939 − 0.342i)47-s + ⋯
L(s)  = 1  + (−0.642 − 0.766i)5-s + (0.939 − 0.342i)7-s + (0.642 − 0.766i)11-s + (0.984 − 0.173i)13-s + (0.5 + 0.866i)17-s + (0.866 + 0.5i)19-s + (−0.939 − 0.342i)23-s + (−0.173 + 0.984i)25-s + (0.984 + 0.173i)29-s + (−0.939 − 0.342i)31-s + (−0.866 − 0.5i)35-s + (0.866 − 0.5i)37-s + (0.173 + 0.984i)41-s + (−0.642 + 0.766i)43-s + (0.939 − 0.342i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.512 - 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.512 - 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.512 - 0.858i$
Analytic conductor: \(46.4248\)
Root analytic conductor: \(46.4248\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 432,\ (1:\ ),\ 0.512 - 0.858i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.980932707 - 1.124564338i\)
\(L(\frac12)\) \(\approx\) \(1.980932707 - 1.124564338i\)
\(L(1)\) \(\approx\) \(1.211513822 - 0.2896743344i\)
\(L(1)\) \(\approx\) \(1.211513822 - 0.2896743344i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-0.642 - 0.766i)T \)
7 \( 1 + (0.939 - 0.342i)T \)
11 \( 1 + (0.642 - 0.766i)T \)
13 \( 1 + (0.984 - 0.173i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 + (-0.939 - 0.342i)T \)
29 \( 1 + (0.984 + 0.173i)T \)
31 \( 1 + (-0.939 - 0.342i)T \)
37 \( 1 + (0.866 - 0.5i)T \)
41 \( 1 + (0.173 + 0.984i)T \)
43 \( 1 + (-0.642 + 0.766i)T \)
47 \( 1 + (0.939 - 0.342i)T \)
53 \( 1 - iT \)
59 \( 1 + (-0.642 - 0.766i)T \)
61 \( 1 + (0.342 + 0.939i)T \)
67 \( 1 + (0.984 - 0.173i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (0.173 - 0.984i)T \)
83 \( 1 + (0.984 + 0.173i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (0.766 + 0.642i)T \)
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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

−23.86255116118190673734004276975, −23.2768389249950725750583006346, −22.32964685803190034452803857847, −21.62024468984979082969261049644, −20.44611770250761451664908868909, −19.91500849084375203891885105798, −18.58519310706240381076164551852, −18.22657732741911820983596408708, −17.277443616288293147464221934273, −15.943717005059403699309580185163, −15.40241455436500506126233416894, −14.31740926249116057672390942595, −13.86533849260820829484477578534, −12.22365728914677526619060816541, −11.66129611609962243835187739251, −10.89350790895619767698999418832, −9.75391479681354457991667527166, −8.682421698276244189219630881532, −7.66975533947247188168480046093, −6.942694970872240923124456151234, −5.72899155906704494589170404046, −4.550405889716600546846412687568, −3.59580872875978166722659969743, −2.37361683545879650734026859801, −1.08147112874609119054971812257, 0.78303090241652457979019029848, 1.58123560490887900817367361257, 3.49696599657863839471382195890, 4.17022338630457517383994429090, 5.34081142490891961647182564412, 6.30445128295832429886984054118, 7.868838378401988138852857911346, 8.20295415457050707498074770659, 9.23867014011052749647048443764, 10.55982887125051531345547960493, 11.425252288269571837415615559373, 12.11994793194772728063065589616, 13.22033693118046051148032569132, 14.1488604885169741825354758411, 14.95316681317051845686043333406, 16.22036900084597382134873860679, 16.56205455661686583687846279852, 17.72640568475646795103576805397, 18.54758771428146483943109462764, 19.65920913132685572027435519231, 20.29380042640514749276304120521, 21.11186279042299784061979736398, 21.937218834506109176827526021520, 23.20768724691216306828781921884, 23.755936460033914457025510302982

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