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Properties

Label 1-3e5-243.229-r0-0-0
Degree $1$
Conductor $243$
Sign $-0.622 - 0.782i$
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  + (0.925 − 0.378i)2-s + (0.713 − 0.700i)4-s + (−0.875 − 0.483i)5-s + (−0.431 − 0.902i)7-s + (0.396 − 0.918i)8-s + (−0.993 − 0.116i)10-s + (−0.981 − 0.192i)11-s + (−0.999 − 0.0387i)13-s + (−0.740 − 0.672i)14-s + (0.0193 − 0.999i)16-s + (0.973 − 0.230i)17-s + (−0.686 + 0.727i)19-s + (−0.963 + 0.268i)20-s + (−0.981 + 0.192i)22-s + (0.996 − 0.0774i)23-s + ⋯
L(s)  = 1  + (0.925 − 0.378i)2-s + (0.713 − 0.700i)4-s + (−0.875 − 0.483i)5-s + (−0.431 − 0.902i)7-s + (0.396 − 0.918i)8-s + (−0.993 − 0.116i)10-s + (−0.981 − 0.192i)11-s + (−0.999 − 0.0387i)13-s + (−0.740 − 0.672i)14-s + (0.0193 − 0.999i)16-s + (0.973 − 0.230i)17-s + (−0.686 + 0.727i)19-s + (−0.963 + 0.268i)20-s + (−0.981 + 0.192i)22-s + (0.996 − 0.0774i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.622 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.622 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(243\)    =    \(3^{5}\)
Sign: $-0.622 - 0.782i$
Analytic conductor: \(1.12848\)
Root analytic conductor: \(1.12848\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{243} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 243,\ (0:\ ),\ -0.622 - 0.782i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6230482609 - 1.292246377i\)
\(L(\frac12)\) \(\approx\) \(0.6230482609 - 1.292246377i\)
\(L(1)\) \(\approx\) \(1.122615621 - 0.7404208348i\)
\(L(1)\) \(\approx\) \(1.122615621 - 0.7404208348i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 + (0.925 - 0.378i)T \)
5 \( 1 + (-0.875 - 0.483i)T \)
7 \( 1 + (-0.431 - 0.902i)T \)
11 \( 1 + (-0.981 - 0.192i)T \)
13 \( 1 + (-0.999 - 0.0387i)T \)
17 \( 1 + (0.973 - 0.230i)T \)
19 \( 1 + (-0.686 + 0.727i)T \)
23 \( 1 + (0.996 - 0.0774i)T \)
29 \( 1 + (-0.211 - 0.977i)T \)
31 \( 1 + (0.0968 - 0.995i)T \)
37 \( 1 + (0.893 + 0.448i)T \)
41 \( 1 + (-0.135 - 0.990i)T \)
43 \( 1 + (0.987 - 0.154i)T \)
47 \( 1 + (0.0968 + 0.995i)T \)
53 \( 1 + (0.766 - 0.642i)T \)
59 \( 1 + (0.323 - 0.946i)T \)
61 \( 1 + (0.713 + 0.700i)T \)
67 \( 1 + (-0.211 + 0.977i)T \)
71 \( 1 + (0.597 - 0.802i)T \)
73 \( 1 + (-0.993 + 0.116i)T \)
79 \( 1 + (-0.790 - 0.612i)T \)
83 \( 1 + (-0.135 + 0.990i)T \)
89 \( 1 + (0.597 + 0.802i)T \)
97 \( 1 + (-0.875 + 0.483i)T \)
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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

−26.20803318598363888094887656829, −25.49924657755543020228050412916, −24.51413647517317479754973700475, −23.47496631898967110043480633943, −23.015469543880481382944449958736, −21.86178536551664566689238303479, −21.35508907088028611245114199034, −19.98651551136415922029587916233, −19.17995180192784752964211872254, −18.12360770470134470867392337879, −16.77475214682277856130426100936, −15.853568466219569519513923539384, −15.0782054444748010914454781210, −14.516701897488000316535127533722, −12.945626030036604009446922639310, −12.41354092490508082197259068227, −11.416490906443383775631798109995, −10.353290009629190265686495884218, −8.73738689891832746210993502802, −7.61475307175321891156805721790, −6.847127320507727153726375225698, −5.54342629486735183686615345238, −4.63214516403520846219532005410, −3.19682145623966189407728048105, −2.51397026657119831377271248005, 0.73879843462215997770429829208, 2.59954772061844532504748489740, 3.75233395062676527336775941174, 4.63673855446514583567281305028, 5.72296825224675407637928969325, 7.18943143369337170272907865933, 7.8988319498519703775486913451, 9.69258536877440990630173796298, 10.57003038232093552379600765279, 11.603427356484127984144400828, 12.62194597208009865829405518651, 13.2086663917014844991426320811, 14.431887895091076157034468851, 15.31372237322355587821485149292, 16.31112819561500245742459368279, 17.01706822561482177690203107546, 18.99971725871757356578913643715, 19.29163597549970248665112006324, 20.644210642276797298378230629040, 20.84621895384638800702187391813, 22.34467177912363861850517840365, 23.14707213567566508138410095567, 23.70417600267750643125488099319, 24.555536183706562017271109069987, 25.68306272449300268548290485382

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