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Properties

Label 1-3e5-243.160-r0-0-0
Degree $1$
Conductor $243$
Sign $0.0258 + 0.999i$
Analytic cond. $1.12848$
Root an. cond. $1.12848$
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  + (0.466 + 0.884i)2-s + (−0.565 + 0.824i)4-s + (0.987 − 0.154i)5-s + (0.813 + 0.581i)7-s + (−0.993 − 0.116i)8-s + (0.597 + 0.802i)10-s + (0.856 − 0.516i)11-s + (−0.740 + 0.672i)13-s + (−0.135 + 0.990i)14-s + (−0.360 − 0.932i)16-s + (−0.286 − 0.957i)17-s + (0.973 − 0.230i)19-s + (−0.431 + 0.902i)20-s + (0.856 + 0.516i)22-s + (0.0968 + 0.995i)23-s + ⋯
L(s)  = 1  + (0.466 + 0.884i)2-s + (−0.565 + 0.824i)4-s + (0.987 − 0.154i)5-s + (0.813 + 0.581i)7-s + (−0.993 − 0.116i)8-s + (0.597 + 0.802i)10-s + (0.856 − 0.516i)11-s + (−0.740 + 0.672i)13-s + (−0.135 + 0.990i)14-s + (−0.360 − 0.932i)16-s + (−0.286 − 0.957i)17-s + (0.973 − 0.230i)19-s + (−0.431 + 0.902i)20-s + (0.856 + 0.516i)22-s + (0.0968 + 0.995i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0258 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0258 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(243\)    =    \(3^{5}\)
Sign: $0.0258 + 0.999i$
Analytic conductor: \(1.12848\)
Root analytic conductor: \(1.12848\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{243} (160, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 243,\ (0:\ ),\ 0.0258 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.278092265 + 1.245464978i\)
\(L(\frac12)\) \(\approx\) \(1.278092265 + 1.245464978i\)
\(L(1)\) \(\approx\) \(1.293469600 + 0.7834030921i\)
\(L(1)\) \(\approx\) \(1.293469600 + 0.7834030921i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 + (0.466 + 0.884i)T \)
5 \( 1 + (0.987 - 0.154i)T \)
7 \( 1 + (0.813 + 0.581i)T \)
11 \( 1 + (0.856 - 0.516i)T \)
13 \( 1 + (-0.740 + 0.672i)T \)
17 \( 1 + (-0.286 - 0.957i)T \)
19 \( 1 + (0.973 - 0.230i)T \)
23 \( 1 + (0.0968 + 0.995i)T \)
29 \( 1 + (-0.790 + 0.612i)T \)
31 \( 1 + (-0.963 + 0.268i)T \)
37 \( 1 + (-0.835 - 0.549i)T \)
41 \( 1 + (0.533 + 0.845i)T \)
43 \( 1 + (-0.981 + 0.192i)T \)
47 \( 1 + (-0.963 - 0.268i)T \)
53 \( 1 + (0.766 + 0.642i)T \)
59 \( 1 + (0.0193 - 0.999i)T \)
61 \( 1 + (-0.565 - 0.824i)T \)
67 \( 1 + (-0.790 - 0.612i)T \)
71 \( 1 + (0.396 - 0.918i)T \)
73 \( 1 + (0.597 - 0.802i)T \)
79 \( 1 + (-0.999 - 0.0387i)T \)
83 \( 1 + (0.533 - 0.845i)T \)
89 \( 1 + (0.396 + 0.918i)T \)
97 \( 1 + (0.987 + 0.154i)T \)
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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

−26.031663190788961541669712940312, −24.61425591000433535236391111937, −24.227783818297649184324561693908, −22.706543159778050587958044214058, −22.28882116574450601731819584015, −21.23412087318982179075410885633, −20.44432492196636834001109521337, −19.75001042889990589918518142848, −18.447891527920965941136933800043, −17.618507349287362509105230887292, −16.928084011371361984673409146362, −14.93920369571389019680120659597, −14.50203157084875559638164648823, −13.51364701759818029224762894217, −12.607306941229256459449800142227, −11.54221799667339610527460890402, −10.465538576334927454640878855370, −9.84202103840150209424282605241, −8.68058408175513900378023142776, −7.12537771161752650287273989606, −5.81974744211438309643353287138, −4.86057253038782798941851247317, −3.73489418362468753556978705419, −2.26905550043633806694433086598, −1.34697687895371319317258522919, 1.77402459245808872103084433654, 3.25680571478939658704309928919, 4.85472474253867872184656039772, 5.43956616176807119729737425212, 6.60700998297098534278457721746, 7.60821643179913048216474314150, 9.080614574025452036820578795527, 9.35573079969869972793927053703, 11.32987137103034509919657653841, 12.15025830609720958582408977937, 13.4317982062995216439168788746, 14.172819323264339961654036026235, 14.821681809064917068546735339914, 16.12376580359342513643826840617, 16.93282737049168171725273968059, 17.8202122963778419364070135480, 18.48491135585087862075829915990, 20.05570001475180138067070721866, 21.31901019504071307439904264345, 21.773036755920207947333274532547, 22.56353938330421955533491768828, 23.969165546080820325257108166643, 24.63711392030980616636372665829, 25.08423049461815509678790994250, 26.23329883065689005245385028024

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