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Properties

Label 1-432-432.83-r0-0-0
Degree $1$
Conductor $432$
Sign $0.858 - 0.512i$
Analytic cond. $2.00619$
Root an. cond. $2.00619$
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  + (0.984 + 0.173i)5-s + (0.766 − 0.642i)7-s + (0.984 − 0.173i)11-s + (−0.342 − 0.939i)13-s + (0.5 − 0.866i)17-s + (−0.866 + 0.5i)19-s + (−0.766 − 0.642i)23-s + (0.939 + 0.342i)25-s + (−0.342 + 0.939i)29-s + (−0.766 − 0.642i)31-s + (0.866 − 0.5i)35-s + (0.866 + 0.5i)37-s + (−0.939 + 0.342i)41-s + (−0.984 + 0.173i)43-s + (0.766 − 0.642i)47-s + ⋯
L(s)  = 1  + (0.984 + 0.173i)5-s + (0.766 − 0.642i)7-s + (0.984 − 0.173i)11-s + (−0.342 − 0.939i)13-s + (0.5 − 0.866i)17-s + (−0.866 + 0.5i)19-s + (−0.766 − 0.642i)23-s + (0.939 + 0.342i)25-s + (−0.342 + 0.939i)29-s + (−0.766 − 0.642i)31-s + (0.866 − 0.5i)35-s + (0.866 + 0.5i)37-s + (−0.939 + 0.342i)41-s + (−0.984 + 0.173i)43-s + (0.766 − 0.642i)47-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.858 - 0.512i$
Analytic conductor: \(2.00619\)
Root analytic conductor: \(2.00619\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 432,\ (0:\ ),\ 0.858 - 0.512i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.678817871 - 0.4629489258i\)
\(L(\frac12)\) \(\approx\) \(1.678817871 - 0.4629489258i\)
\(L(1)\) \(\approx\) \(1.347401966 - 0.1692044228i\)
\(L(1)\) \(\approx\) \(1.347401966 - 0.1692044228i\)

Euler product

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

−24.21247268532702795365486895744, −23.57720834791569648592840976377, −22.08604583145828851909413921871, −21.686560682980983197371256142327, −21.01432468710950282805238235635, −19.8847619532080991665659645993, −18.996517627369934472760114116674, −18.03629527613889410417120543077, −17.21666731999304954603880254293, −16.6966315129434087994235289224, −15.26459162963034730918255093128, −14.50445040444890987644081441148, −13.831489940225910147975962640435, −12.65796408421537950573551681111, −11.85955600205296589156711106443, −10.90439780794163070143981087111, −9.69993429667834691868058469433, −9.05863408351462756547764777617, −8.101719116872399940135822779471, −6.73247613077974798530962368635, −5.922646019803426904442759668844, −4.92143463257522054250177478535, −3.86059614831762520005079911097, −2.15554901985904573257200923141, −1.6275742490544422260232927093, 1.12674403236168949338801664885, 2.2201039553281804838591621958, 3.53891414858091229844001756948, 4.746304259329585908627090770458, 5.72535846147039389988852593162, 6.71796276412638513488023758876, 7.74798339061125318053129615224, 8.79493713075795126923915714837, 9.88840534866949903709819031690, 10.55530165015781746908494754348, 11.576471450986899028801229596981, 12.65101133918142082008356028587, 13.64932288223461534706422769184, 14.41807902887612705822123894654, 14.97904411416999958450429935873, 16.676619695328149130864479318361, 16.95104470686487589362284600771, 18.0347034904708524851060949489, 18.61142619965800657263427104444, 20.11433755735475883552779948898, 20.43427169915040375241722494744, 21.60203846423727200498161479736, 22.204007080610162873159555929354, 23.15919366900622282143452306092, 24.15348036033255833280440857690

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