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Properties

Label 1-3e5-243.47-r1-0-0
Degree $1$
Conductor $243$
Sign $0.522 + 0.852i$
Analytic cond. $26.1139$
Root an. cond. $26.1139$
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  + (−0.713 + 0.700i)2-s + (0.0193 − 0.999i)4-s + (−0.533 − 0.845i)5-s + (−0.627 + 0.778i)7-s + (0.686 + 0.727i)8-s + (0.973 + 0.230i)10-s + (−0.925 − 0.378i)11-s + (0.996 + 0.0774i)13-s + (−0.0968 − 0.995i)14-s + (−0.999 − 0.0387i)16-s + (−0.893 + 0.448i)17-s + (−0.0581 − 0.998i)19-s + (−0.856 + 0.516i)20-s + (0.925 − 0.378i)22-s + (−0.987 + 0.154i)23-s + ⋯
L(s)  = 1  + (−0.713 + 0.700i)2-s + (0.0193 − 0.999i)4-s + (−0.533 − 0.845i)5-s + (−0.627 + 0.778i)7-s + (0.686 + 0.727i)8-s + (0.973 + 0.230i)10-s + (−0.925 − 0.378i)11-s + (0.996 + 0.0774i)13-s + (−0.0968 − 0.995i)14-s + (−0.999 − 0.0387i)16-s + (−0.893 + 0.448i)17-s + (−0.0581 − 0.998i)19-s + (−0.856 + 0.516i)20-s + (0.925 − 0.378i)22-s + (−0.987 + 0.154i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(243\)    =    \(3^{5}\)
Sign: $0.522 + 0.852i$
Analytic conductor: \(26.1139\)
Root analytic conductor: \(26.1139\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{243} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 243,\ (1:\ ),\ 0.522 + 0.852i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6363065098 + 0.3564838604i\)
\(L(\frac12)\) \(\approx\) \(0.6363065098 + 0.3564838604i\)
\(L(1)\) \(\approx\) \(0.5884481521 + 0.1284568845i\)
\(L(1)\) \(\approx\) \(0.5884481521 + 0.1284568845i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 + (-0.713 + 0.700i)T \)
5 \( 1 + (-0.533 - 0.845i)T \)
7 \( 1 + (-0.627 + 0.778i)T \)
11 \( 1 + (-0.925 - 0.378i)T \)
13 \( 1 + (0.996 + 0.0774i)T \)
17 \( 1 + (-0.893 + 0.448i)T \)
19 \( 1 + (-0.0581 - 0.998i)T \)
23 \( 1 + (-0.987 + 0.154i)T \)
29 \( 1 + (0.910 - 0.413i)T \)
31 \( 1 + (-0.981 - 0.192i)T \)
37 \( 1 + (0.597 + 0.802i)T \)
41 \( 1 + (0.963 - 0.268i)T \)
43 \( 1 + (0.952 - 0.305i)T \)
47 \( 1 + (0.981 - 0.192i)T \)
53 \( 1 + (-0.173 + 0.984i)T \)
59 \( 1 + (0.790 + 0.612i)T \)
61 \( 1 + (0.0193 + 0.999i)T \)
67 \( 1 + (-0.910 - 0.413i)T \)
71 \( 1 + (0.286 + 0.957i)T \)
73 \( 1 + (0.973 - 0.230i)T \)
79 \( 1 + (0.249 + 0.968i)T \)
83 \( 1 + (0.963 + 0.268i)T \)
89 \( 1 + (0.286 - 0.957i)T \)
97 \( 1 + (0.533 - 0.845i)T \)
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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

−26.053131687655367124283858437696, −25.32531533347990878262891188979, −23.63350644238374572955911678049, −22.89590486673133781288271784454, −22.09086295727061706482948724468, −20.85634214977916497474767276941, −20.10533516132515377637080559938, −19.28568689927674711342278958656, −18.28053177251627506601003301406, −17.798776310676023838652012339656, −16.18997029516986480726735815736, −15.89087978019894270663352883424, −14.25019775566296224964147792696, −13.20034374530197000792921540263, −12.27728807858053607494911069080, −10.949825080764597412735826676354, −10.57148964986450090089472867586, −9.505408698192931399364798885145, −8.13133008936438807053329313660, −7.368898096217028074810765884076, −6.31177500150350457851343894411, −4.2279944182501367946725162121, −3.39326653198343290591826768753, −2.228994382742102201097059780862, −0.47286003824804978147021894112, 0.71854766074969500173771141109, 2.38940010005339022307696484065, 4.20171734561661717198789787706, 5.48419373576362688991836799422, 6.29352847200299477857335367393, 7.678178281151777451129289270604, 8.639598057256499047264907618457, 9.16937840880988493828363526490, 10.53759122682392755494342520775, 11.54862535367842454220992208459, 12.87923298491256106517784414283, 13.691506542064644349778413394468, 15.3778435242986395036787813839, 15.7239297539974275164596190479, 16.46707967262767331530806342486, 17.693814586725275314131249254356, 18.52772675921906643211265256462, 19.4482727991428079184777625116, 20.19783227080082822061019153449, 21.38556394970740728227591806507, 22.60318322002855683769763779621, 23.83346929899031478949118931979, 24.05635098891245266381216914284, 25.32185700261937795273585481388, 25.95922403466125476004735267283

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