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Properties

Label 1-3e5-243.16-r0-0-0
Degree $1$
Conductor $243$
Sign $0.950 - 0.311i$
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  + (0.987 + 0.154i)2-s + (0.952 + 0.305i)4-s + (−0.910 − 0.413i)5-s + (−0.135 − 0.990i)7-s + (0.893 + 0.448i)8-s + (−0.835 − 0.549i)10-s + (0.996 − 0.0774i)11-s + (0.323 + 0.946i)13-s + (0.0193 − 0.999i)14-s + (0.813 + 0.581i)16-s + (0.396 − 0.918i)17-s + (0.597 − 0.802i)19-s + (−0.740 − 0.672i)20-s + (0.996 + 0.0774i)22-s + (−0.790 − 0.612i)23-s + ⋯
L(s)  = 1  + (0.987 + 0.154i)2-s + (0.952 + 0.305i)4-s + (−0.910 − 0.413i)5-s + (−0.135 − 0.990i)7-s + (0.893 + 0.448i)8-s + (−0.835 − 0.549i)10-s + (0.996 − 0.0774i)11-s + (0.323 + 0.946i)13-s + (0.0193 − 0.999i)14-s + (0.813 + 0.581i)16-s + (0.396 − 0.918i)17-s + (0.597 − 0.802i)19-s + (−0.740 − 0.672i)20-s + (0.996 + 0.0774i)22-s + (−0.790 − 0.612i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.950 - 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.950 - 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(243\)    =    \(3^{5}\)
Sign: $0.950 - 0.311i$
Analytic conductor: \(1.12848\)
Root analytic conductor: \(1.12848\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{243} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 243,\ (0:\ ),\ 0.950 - 0.311i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.008146996 - 0.3207211177i\)
\(L(\frac12)\) \(\approx\) \(2.008146996 - 0.3207211177i\)
\(L(1)\) \(\approx\) \(1.716913867 - 0.1055989782i\)
\(L(1)\) \(\approx\) \(1.716913867 - 0.1055989782i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 + (0.987 + 0.154i)T \)
5 \( 1 + (-0.910 - 0.413i)T \)
7 \( 1 + (-0.135 - 0.990i)T \)
11 \( 1 + (0.996 - 0.0774i)T \)
13 \( 1 + (0.323 + 0.946i)T \)
17 \( 1 + (0.396 - 0.918i)T \)
19 \( 1 + (0.597 - 0.802i)T \)
23 \( 1 + (-0.790 - 0.612i)T \)
29 \( 1 + (0.856 - 0.516i)T \)
31 \( 1 + (-0.999 + 0.0387i)T \)
37 \( 1 + (-0.686 + 0.727i)T \)
41 \( 1 + (-0.360 + 0.932i)T \)
43 \( 1 + (0.249 + 0.968i)T \)
47 \( 1 + (-0.999 - 0.0387i)T \)
53 \( 1 + (-0.939 + 0.342i)T \)
59 \( 1 + (-0.431 - 0.902i)T \)
61 \( 1 + (0.952 - 0.305i)T \)
67 \( 1 + (0.856 + 0.516i)T \)
71 \( 1 + (-0.0581 + 0.998i)T \)
73 \( 1 + (-0.835 + 0.549i)T \)
79 \( 1 + (-0.627 + 0.778i)T \)
83 \( 1 + (-0.360 - 0.932i)T \)
89 \( 1 + (-0.0581 - 0.998i)T \)
97 \( 1 + (-0.910 + 0.413i)T \)
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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

−25.84343814426455174798774598891, −25.15488859367877621207021578295, −24.20480065536161924096324655929, −23.297799018157871036499447761249, −22.39801969136839444847748011061, −21.93816604635580906613541288922, −20.70186720867454710306882680240, −19.71852885807325323507722574225, −19.101217503760042431633287389449, −17.916209733550606743146105839039, −16.39066989039189518617810416263, −15.59254581744895154699115975906, −14.84939973726955893129658769529, −14.05458304081088092888316543103, −12.51198902764819482686679036996, −12.139032485979726089991622347084, −11.12100234335087607410324808099, −10.07133769324462720991228400448, −8.5111036008968917934470356825, −7.42335036322120955945240632506, −6.22583115460577041756922650850, −5.374256206183305580946998600770, −3.83798021357323246358259348876, −3.25382085455387636312092628085, −1.71419735567996586059265658527, 1.28820353367725006173843007725, 3.18063595510977837570176296520, 4.132266813661574091827201785017, 4.84051376103164606013248728576, 6.498098398729538913776419432209, 7.19311033563497329485038580566, 8.30594554916427182514368588880, 9.72461907207115002493957176404, 11.28624685742979308157262302569, 11.68921581525924125748979722804, 12.82827092491522954417461566876, 13.908009640349182206236414644247, 14.499559290184031879916060575089, 15.94236955913506660568837782982, 16.33037660244154895261257136414, 17.31248323251533758904234849917, 18.97621087470325109508335752253, 20.012690390187257500022570710170, 20.37983587063296105638592093625, 21.62031701472213722404532048043, 22.630277582650602605245756660859, 23.33998803402747470949634444132, 24.09023279664071338604379183422, 24.832455079044080344512323124780, 26.07779728665933184333293547719

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