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Properties

Label 1-3e5-243.49-r0-0-0
Degree $1$
Conductor $243$
Sign $-0.786 - 0.617i$
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  + (0.657 − 0.753i)2-s + (−0.135 − 0.990i)4-s + (0.713 − 0.700i)5-s + (−0.999 + 0.0387i)7-s + (−0.835 − 0.549i)8-s + (−0.0581 − 0.998i)10-s + (−0.910 − 0.413i)11-s + (0.856 − 0.516i)13-s + (−0.627 + 0.778i)14-s + (−0.963 + 0.268i)16-s + (−0.993 − 0.116i)17-s + (0.396 − 0.918i)19-s + (−0.790 − 0.612i)20-s + (−0.910 + 0.413i)22-s + (0.466 + 0.884i)23-s + ⋯
L(s)  = 1  + (0.657 − 0.753i)2-s + (−0.135 − 0.990i)4-s + (0.713 − 0.700i)5-s + (−0.999 + 0.0387i)7-s + (−0.835 − 0.549i)8-s + (−0.0581 − 0.998i)10-s + (−0.910 − 0.413i)11-s + (0.856 − 0.516i)13-s + (−0.627 + 0.778i)14-s + (−0.963 + 0.268i)16-s + (−0.993 − 0.116i)17-s + (0.396 − 0.918i)19-s + (−0.790 − 0.612i)20-s + (−0.910 + 0.413i)22-s + (0.466 + 0.884i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.786 - 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.786 - 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(243\)    =    \(3^{5}\)
Sign: $-0.786 - 0.617i$
Analytic conductor: \(1.12848\)
Root analytic conductor: \(1.12848\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{243} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 243,\ (0:\ ),\ -0.786 - 0.617i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4830085986 - 1.396889512i\)
\(L(\frac12)\) \(\approx\) \(0.4830085986 - 1.396889512i\)
\(L(1)\) \(\approx\) \(0.9962145826 - 0.8951007490i\)
\(L(1)\) \(\approx\) \(0.9962145826 - 0.8951007490i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 + (0.657 - 0.753i)T \)
5 \( 1 + (0.713 - 0.700i)T \)
7 \( 1 + (-0.999 + 0.0387i)T \)
11 \( 1 + (-0.910 - 0.413i)T \)
13 \( 1 + (0.856 - 0.516i)T \)
17 \( 1 + (-0.993 - 0.116i)T \)
19 \( 1 + (0.396 - 0.918i)T \)
23 \( 1 + (0.466 + 0.884i)T \)
29 \( 1 + (0.987 - 0.154i)T \)
31 \( 1 + (-0.211 + 0.977i)T \)
37 \( 1 + (0.973 - 0.230i)T \)
41 \( 1 + (0.323 - 0.946i)T \)
43 \( 1 + (-0.565 + 0.824i)T \)
47 \( 1 + (-0.211 - 0.977i)T \)
53 \( 1 + (-0.939 - 0.342i)T \)
59 \( 1 + (0.0968 - 0.995i)T \)
61 \( 1 + (-0.135 + 0.990i)T \)
67 \( 1 + (0.987 + 0.154i)T \)
71 \( 1 + (0.893 + 0.448i)T \)
73 \( 1 + (-0.0581 + 0.998i)T \)
79 \( 1 + (-0.981 - 0.192i)T \)
83 \( 1 + (0.323 + 0.946i)T \)
89 \( 1 + (0.893 - 0.448i)T \)
97 \( 1 + (0.713 + 0.700i)T \)
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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

−26.222701880930870747222481570081, −25.59580864219818393707257004657, −24.82496939911266575241301361302, −23.56132027118017095303457092301, −22.85587205396800004986067797300, −22.13552836699695563315711950619, −21.221224752867056426925246865224, −20.316863103057226775975824209078, −18.695073909291280841421451005476, −18.12611469098616133225487476695, −16.94756173302213648353689777399, −16.04959378209213732840498536875, −15.233781637785953687623170393295, −14.162836821704622084648867679520, −13.31638254127878737640998752459, −12.67340240412396065907883547159, −11.21761363793311936350401422684, −10.070750263583049065459960427408, −8.984931792329398361791331665239, −7.68042424545613201887834994369, −6.504749353719393402125033437949, −6.08013426450784575060122716098, −4.65746368775870100991029607728, −3.35868878397402549168871363810, −2.36277335429512723979126782823, 0.84726095800792416477079865107, 2.422831241371292051372998903917, 3.393218911671358898877189286121, 4.84827425488305385398267983571, 5.70629461314344671805116834174, 6.68497053055542810256432688661, 8.604795550167222351249555101131, 9.486446114374704308758282102427, 10.42931087415741818300601439235, 11.40588238008278673539920197945, 12.83927916885785371809279225161, 13.16785345013824615090228786230, 13.937816447034204624292803319947, 15.60561444797805287204886045713, 15.99361468570230859129209214291, 17.60145314394211304957649794465, 18.40877082159915324974932953437, 19.62974862327721721609968317907, 20.23756953583249185652325355205, 21.31893755533309576072284498163, 21.85356667605930158402213422776, 22.987476395160346149371737513861, 23.7279209404472051636530603402, 24.770707289219682568460172813377, 25.61902901316111498902691990130

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