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Properties

Label 1-432-432.389-r1-0-0
Degree $1$
Conductor $432$
Sign $-0.969 - 0.244i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.969 - 0.244i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.969 - 0.244i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.969 - 0.244i$
Analytic conductor: \(46.4248\)
Root analytic conductor: \(46.4248\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (389, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 432,\ (1:\ ),\ -0.969 - 0.244i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1029047645 - 0.8281327524i\)
\(L(\frac12)\) \(\approx\) \(0.1029047645 - 0.8281327524i\)
\(L(1)\) \(\approx\) \(0.9602816711 - 0.2508458919i\)
\(L(1)\) \(\approx\) \(0.9602816711 - 0.2508458919i\)

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

−24.13078759682021599686930380288, −23.60960266912087475669620568873, −22.462557719592457843735481837746, −21.733862053638536853250932898684, −20.96729664355126936676968998784, −20.10848273857375062147512879977, −19.00717153803244284939845410311, −18.17162540404147334433410203144, −17.40150758098115655304960265736, −16.80511583868691627554344642256, −15.17782984565827901097624225892, −14.7916189598318339377768579197, −13.9036186424936205700864238264, −12.89681397902333022067768186311, −11.90874715908880666150648941661, −10.69088770306922112200812783859, −10.28622336587916677727980065390, −9.15684898073053411379490598118, −7.82450346485677914651989705373, −7.21224587283998772335602289551, −6.011494812808105133767659813810, −4.99996664930771521276835455754, −3.95928511918960550456747273815, −2.426931867918678573105661540294, −1.76887195132218477982699776043, 0.19685093741208107325903704027, 1.63988878868623664123769555944, 2.56454088372888125920662253018, 4.13393784755808769273569108492, 5.37183428796783267433573727479, 5.62856025237840230151605521091, 7.328271566305393365732944810404, 8.21826638933662088240120929671, 9.06482080149259311397671676767, 10.05374598966283312469121711509, 11.04517517889030975470221021903, 12.12625255850300888459379164968, 12.83067683657365331818998051417, 13.974292924054084947748426910800, 14.54439997068643400395246768153, 15.78508152681074303060604079715, 16.60981138264825162027706365005, 17.46426700358773339757268103201, 18.210114085153170523166083317032, 19.14573001776646947112259933861, 20.30127827480794458681790615169, 21.02190957125786276271620722772, 21.58655045282521803164860552598, 22.53857417425469057312185657155, 23.95175380339397253148280044224

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