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Properties

Label 1-3e5-243.148-r0-0-0
Degree $1$
Conductor $243$
Sign $0.988 - 0.148i$
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  + (−0.790 − 0.612i)2-s + (0.249 + 0.968i)4-s + (0.856 − 0.516i)5-s + (−0.565 + 0.824i)7-s + (0.396 − 0.918i)8-s + (−0.993 − 0.116i)10-s + (0.323 + 0.946i)11-s + (0.533 − 0.845i)13-s + (0.952 − 0.305i)14-s + (−0.875 + 0.483i)16-s + (0.973 − 0.230i)17-s + (−0.686 + 0.727i)19-s + (0.713 + 0.700i)20-s + (0.323 − 0.946i)22-s + (−0.431 + 0.902i)23-s + ⋯
L(s)  = 1  + (−0.790 − 0.612i)2-s + (0.249 + 0.968i)4-s + (0.856 − 0.516i)5-s + (−0.565 + 0.824i)7-s + (0.396 − 0.918i)8-s + (−0.993 − 0.116i)10-s + (0.323 + 0.946i)11-s + (0.533 − 0.845i)13-s + (0.952 − 0.305i)14-s + (−0.875 + 0.483i)16-s + (0.973 − 0.230i)17-s + (−0.686 + 0.727i)19-s + (0.713 + 0.700i)20-s + (0.323 − 0.946i)22-s + (−0.431 + 0.902i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(243\)    =    \(3^{5}\)
Sign: $0.988 - 0.148i$
Analytic conductor: \(1.12848\)
Root analytic conductor: \(1.12848\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{243} (148, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 243,\ (0:\ ),\ 0.988 - 0.148i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9294253911 - 0.06921926801i\)
\(L(\frac12)\) \(\approx\) \(0.9294253911 - 0.06921926801i\)
\(L(1)\) \(\approx\) \(0.8418723534 - 0.1197103576i\)
\(L(1)\) \(\approx\) \(0.8418723534 - 0.1197103576i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 + (-0.790 - 0.612i)T \)
5 \( 1 + (0.856 - 0.516i)T \)
7 \( 1 + (-0.565 + 0.824i)T \)
11 \( 1 + (0.323 + 0.946i)T \)
13 \( 1 + (0.533 - 0.845i)T \)
17 \( 1 + (0.973 - 0.230i)T \)
19 \( 1 + (-0.686 + 0.727i)T \)
23 \( 1 + (-0.431 + 0.902i)T \)
29 \( 1 + (-0.740 + 0.672i)T \)
31 \( 1 + (0.813 + 0.581i)T \)
37 \( 1 + (0.893 + 0.448i)T \)
41 \( 1 + (0.925 + 0.378i)T \)
43 \( 1 + (-0.627 - 0.778i)T \)
47 \( 1 + (0.813 - 0.581i)T \)
53 \( 1 + (0.766 - 0.642i)T \)
59 \( 1 + (0.657 + 0.753i)T \)
61 \( 1 + (0.249 - 0.968i)T \)
67 \( 1 + (-0.740 - 0.672i)T \)
71 \( 1 + (0.597 - 0.802i)T \)
73 \( 1 + (-0.993 + 0.116i)T \)
79 \( 1 + (-0.135 + 0.990i)T \)
83 \( 1 + (0.925 - 0.378i)T \)
89 \( 1 + (0.597 + 0.802i)T \)
97 \( 1 + (0.856 + 0.516i)T \)
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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

−26.307177993648586370060692555770, −25.48530662220866450152041756649, −24.472989592162294627947579271, −23.57613748874529395740182254007, −22.67434098941952016814669381906, −21.51081322731578781333521442299, −20.5248851166542434791882665751, −19.19663742107823460164800156897, −18.835474873380954723422203350484, −17.61729061904973370310396223156, −16.77478336863243259053749697072, −16.22156846819294372767013295400, −14.78703672817435866606696580674, −14.01470401981088908734744805210, −13.23646361460989110596589164142, −11.37516285602341664984964592844, −10.53713672395504255012647746539, −9.68172788738484504653177029597, −8.76261220658653365632493793965, −7.48626245736048639849285977501, −6.39056216719039474164454518392, −5.93369484401817332852754036382, −4.14727326866508354190693216431, −2.531904121625202333398917958121, −1.02319512886782196679413087691, 1.34040721876394963601405698177, 2.42878916180180005347396618824, 3.68688435868735955183315090398, 5.34029345553889369100459457630, 6.43023229003396575030896462810, 7.84715301077801741940414803923, 8.88366453534017950781049362244, 9.72556175905480986820101027805, 10.375788896966898063112045928989, 11.9295459971728976505854618438, 12.56978497607820936707512388897, 13.40683154908047941498105516858, 14.930565609129641685140781920101, 16.08297572095367099900650845328, 16.94054990075366221760356142304, 17.88258548474980318433641198005, 18.545404454713709394440082173582, 19.67231455336694659579167327049, 20.541456571300872054379501690225, 21.3115526943638019552196527575, 22.16507682483751037553361628194, 23.1575905056762805403971021920, 24.8349388616607923742458373083, 25.46913494957470564420870975752, 25.781133371504572217469022508355

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