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Properties

Label 1-3e5-243.70-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: $\chi_{243} (70, \cdot )$
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

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

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