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Properties

Label 1-3e5-243.184-r0-0-0
Degree $1$
Conductor $243$
Sign $-0.999 + 0.0323i$
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  + (−0.875 + 0.483i)2-s + (0.533 − 0.845i)4-s + (−0.565 − 0.824i)5-s + (−0.740 + 0.672i)7-s + (−0.0581 + 0.998i)8-s + (0.893 + 0.448i)10-s + (0.249 − 0.968i)11-s + (−0.627 + 0.778i)13-s + (0.323 − 0.946i)14-s + (−0.431 − 0.902i)16-s + (0.597 − 0.802i)17-s + (−0.993 + 0.116i)19-s + (−0.999 + 0.0387i)20-s + (0.249 + 0.968i)22-s + (−0.211 + 0.977i)23-s + ⋯
L(s)  = 1  + (−0.875 + 0.483i)2-s + (0.533 − 0.845i)4-s + (−0.565 − 0.824i)5-s + (−0.740 + 0.672i)7-s + (−0.0581 + 0.998i)8-s + (0.893 + 0.448i)10-s + (0.249 − 0.968i)11-s + (−0.627 + 0.778i)13-s + (0.323 − 0.946i)14-s + (−0.431 − 0.902i)16-s + (0.597 − 0.802i)17-s + (−0.993 + 0.116i)19-s + (−0.999 + 0.0387i)20-s + (0.249 + 0.968i)22-s + (−0.211 + 0.977i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0323i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0323i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(243\)    =    \(3^{5}\)
Sign: $-0.999 + 0.0323i$
Analytic conductor: \(1.12848\)
Root analytic conductor: \(1.12848\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{243} (184, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 243,\ (0:\ ),\ -0.999 + 0.0323i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.0008819556334 + 0.05457017133i\)
\(L(\frac12)\) \(\approx\) \(0.0008819556334 + 0.05457017133i\)
\(L(1)\) \(\approx\) \(0.4449108151 + 0.05283980134i\)
\(L(1)\) \(\approx\) \(0.4449108151 + 0.05283980134i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 + (-0.875 + 0.483i)T \)
5 \( 1 + (-0.565 - 0.824i)T \)
7 \( 1 + (-0.740 + 0.672i)T \)
11 \( 1 + (0.249 - 0.968i)T \)
13 \( 1 + (-0.627 + 0.778i)T \)
17 \( 1 + (0.597 - 0.802i)T \)
19 \( 1 + (-0.993 + 0.116i)T \)
23 \( 1 + (-0.211 + 0.977i)T \)
29 \( 1 + (-0.981 - 0.192i)T \)
31 \( 1 + (-0.790 + 0.612i)T \)
37 \( 1 + (-0.286 + 0.957i)T \)
41 \( 1 + (0.0193 + 0.999i)T \)
43 \( 1 + (-0.910 - 0.413i)T \)
47 \( 1 + (-0.790 - 0.612i)T \)
53 \( 1 + (-0.939 - 0.342i)T \)
59 \( 1 + (-0.963 - 0.268i)T \)
61 \( 1 + (0.533 + 0.845i)T \)
67 \( 1 + (-0.981 + 0.192i)T \)
71 \( 1 + (-0.835 + 0.549i)T \)
73 \( 1 + (0.893 - 0.448i)T \)
79 \( 1 + (0.856 + 0.516i)T \)
83 \( 1 + (0.0193 - 0.999i)T \)
89 \( 1 + (-0.835 - 0.549i)T \)
97 \( 1 + (-0.565 + 0.824i)T \)
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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

−25.979044444365844197752196684889, −25.19250292672578653862952916814, −23.777643230820774825056341172412, −22.68269704964716172984434282476, −22.13229869215008891024556496655, −20.75436139393055600456744333959, −19.85087371906629284323263955289, −19.30987333392776245124878098109, −18.35222218069623551340687205669, −17.325220822327560808597272347538, −16.57826586262923033128433178111, −15.36634005015768134428205739486, −14.580246612424780937922238586843, −12.84162583347195054693699251873, −12.33761206886412569486587897127, −10.901099786415134353665498007016, −10.3511994206964572582300537148, −9.43688059979835521258927828817, −7.97909723882244120343483353772, −7.25418377376952632691245321178, −6.33848226697083310280453245570, −4.18205227270663466250491108408, −3.28331115702045804272005585218, −2.04604022343768194205714699167, −0.049192508819544847016837502352, 1.67681929695494250513081900421, 3.31481657148777274476328531310, 4.982145939103629678458495000929, 5.99184615260532573533121802853, 7.12327312908269549940763315610, 8.27277625142643604331152477887, 9.08502906073762087489437607383, 9.811057522566919917511575081701, 11.35664642560028715153495205555, 12.0245774630045068159767853043, 13.33394184824771940132517723115, 14.63363154434326395507387702023, 15.60341774270368887647272764468, 16.512415376383483805838549466994, 16.85648726877207317339405615241, 18.37291893287988582896513361021, 19.19640863238847953870852933746, 19.685620403999342906332496408038, 20.899812749126430270397214192834, 21.93708176165282647187946437284, 23.30927985001923342768529376734, 24.01311960043220632919395956252, 24.89326554897833563068004827162, 25.589462613107199995075963649248, 26.710295980771501667378099496520

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