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Properties

Label 1-3e5-243.13-r0-0-0
Degree $1$
Conductor $243$
Sign $0.317 + 0.948i$
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  + (0.952 + 0.305i)2-s + (0.813 + 0.581i)4-s + (0.657 + 0.753i)5-s + (−0.963 + 0.268i)7-s + (0.597 + 0.802i)8-s + (0.396 + 0.918i)10-s + (0.987 − 0.154i)11-s + (−0.790 + 0.612i)13-s + (−0.999 − 0.0387i)14-s + (0.323 + 0.946i)16-s + (−0.686 − 0.727i)17-s + (−0.286 − 0.957i)19-s + (0.0968 + 0.995i)20-s + (0.987 + 0.154i)22-s + (0.249 + 0.968i)23-s + ⋯
L(s)  = 1  + (0.952 + 0.305i)2-s + (0.813 + 0.581i)4-s + (0.657 + 0.753i)5-s + (−0.963 + 0.268i)7-s + (0.597 + 0.802i)8-s + (0.396 + 0.918i)10-s + (0.987 − 0.154i)11-s + (−0.790 + 0.612i)13-s + (−0.999 − 0.0387i)14-s + (0.323 + 0.946i)16-s + (−0.686 − 0.727i)17-s + (−0.286 − 0.957i)19-s + (0.0968 + 0.995i)20-s + (0.987 + 0.154i)22-s + (0.249 + 0.968i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.317 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.317 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(243\)    =    \(3^{5}\)
Sign: $0.317 + 0.948i$
Analytic conductor: \(1.12848\)
Root analytic conductor: \(1.12848\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{243} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 243,\ (0:\ ),\ 0.317 + 0.948i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.764507723 + 1.269830913i\)
\(L(\frac12)\) \(\approx\) \(1.764507723 + 1.269830913i\)
\(L(1)\) \(\approx\) \(1.684534566 + 0.6972689120i\)
\(L(1)\) \(\approx\) \(1.684534566 + 0.6972689120i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 + (0.952 + 0.305i)T \)
5 \( 1 + (0.657 + 0.753i)T \)
7 \( 1 + (-0.963 + 0.268i)T \)
11 \( 1 + (0.987 - 0.154i)T \)
13 \( 1 + (-0.790 + 0.612i)T \)
17 \( 1 + (-0.686 - 0.727i)T \)
19 \( 1 + (-0.286 - 0.957i)T \)
23 \( 1 + (0.249 + 0.968i)T \)
29 \( 1 + (0.466 - 0.884i)T \)
31 \( 1 + (0.996 - 0.0774i)T \)
37 \( 1 + (-0.0581 - 0.998i)T \)
41 \( 1 + (-0.740 - 0.672i)T \)
43 \( 1 + (-0.875 + 0.483i)T \)
47 \( 1 + (0.996 + 0.0774i)T \)
53 \( 1 + (0.766 - 0.642i)T \)
59 \( 1 + (-0.627 + 0.778i)T \)
61 \( 1 + (0.813 - 0.581i)T \)
67 \( 1 + (0.466 + 0.884i)T \)
71 \( 1 + (-0.993 - 0.116i)T \)
73 \( 1 + (0.396 - 0.918i)T \)
79 \( 1 + (-0.211 - 0.977i)T \)
83 \( 1 + (-0.740 + 0.672i)T \)
89 \( 1 + (-0.993 + 0.116i)T \)
97 \( 1 + (0.657 - 0.753i)T \)
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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

−25.48507312559724259976715577144, −24.98557047228148764981800589531, −24.13308495162659072653029328762, −23.02230220440475584196111822897, −22.211916110044825252225075853321, −21.56994540934597847483403750058, −20.25334604007534127862029354333, −19.943786430391958520347593537356, −18.81153702186225718401376958541, −17.16190344436822075100362328603, −16.63793261911535177460097002947, −15.439078356063657186173522458308, −14.4407621473378709773136545446, −13.47490507563607213866176152933, −12.62517816575392792531008012143, −12.06972464773665724481130913916, −10.43013709960620250836060559454, −9.84980762967059940030835182446, −8.54465492799108253277669748375, −6.804259401913077693190122781154, −6.13944887834516359219039722545, −4.91552553011446692046849015030, −3.91827742527935088365996229954, −2.619895933044010166520951759202, −1.28066237405150729743852789018, 2.18640865961725466266758316972, 3.05870150956724122537446127586, 4.30905183896424348064957498298, 5.63514043175258058289987552419, 6.6729276189954691605979643952, 7.0864345474827882257482387326, 8.967395783600543673098578808479, 9.91644883966651920407199288536, 11.2894579986332566887366892398, 12.04961775249212724481589096995, 13.35669198660742656763680288593, 13.87673275669089261550280772469, 14.960015156951659843719206804112, 15.73888074205655191671693910187, 16.91084795549380753790569444914, 17.637342221523027168775626741, 19.16827440233501251942353236309, 19.768551981022054978546317159416, 21.215004067514427206181264022202, 21.99336479455160636512733241609, 22.435287772600505657192692791122, 23.39450388272928179421454788286, 24.680640547574134000478568705245, 25.16482657729033229474026576337, 26.167509611745694613756927438249

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