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Properties

Label 1-3e5-243.43-r0-0-0
Degree $1$
Conductor $243$
Sign $0.928 + 0.372i$
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  + (0.323 − 0.946i)2-s + (−0.790 − 0.612i)4-s + (−0.963 + 0.268i)5-s + (0.466 + 0.884i)7-s + (−0.835 + 0.549i)8-s + (−0.0581 + 0.998i)10-s + (0.813 + 0.581i)11-s + (−0.875 + 0.483i)13-s + (0.987 − 0.154i)14-s + (0.249 + 0.968i)16-s + (−0.993 + 0.116i)17-s + (0.396 + 0.918i)19-s + (0.925 + 0.378i)20-s + (0.813 − 0.581i)22-s + (0.533 + 0.845i)23-s + ⋯
L(s)  = 1  + (0.323 − 0.946i)2-s + (−0.790 − 0.612i)4-s + (−0.963 + 0.268i)5-s + (0.466 + 0.884i)7-s + (−0.835 + 0.549i)8-s + (−0.0581 + 0.998i)10-s + (0.813 + 0.581i)11-s + (−0.875 + 0.483i)13-s + (0.987 − 0.154i)14-s + (0.249 + 0.968i)16-s + (−0.993 + 0.116i)17-s + (0.396 + 0.918i)19-s + (0.925 + 0.378i)20-s + (0.813 − 0.581i)22-s + (0.533 + 0.845i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(243\)    =    \(3^{5}\)
Sign: $0.928 + 0.372i$
Analytic conductor: \(1.12848\)
Root analytic conductor: \(1.12848\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{243} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 243,\ (0:\ ),\ 0.928 + 0.372i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9155690204 + 0.1767404922i\)
\(L(\frac12)\) \(\approx\) \(0.9155690204 + 0.1767404922i\)
\(L(1)\) \(\approx\) \(0.9286916088 - 0.1580230383i\)
\(L(1)\) \(\approx\) \(0.9286916088 - 0.1580230383i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 + (0.323 - 0.946i)T \)
5 \( 1 + (-0.963 + 0.268i)T \)
7 \( 1 + (0.466 + 0.884i)T \)
11 \( 1 + (0.813 + 0.581i)T \)
13 \( 1 + (-0.875 + 0.483i)T \)
17 \( 1 + (-0.993 + 0.116i)T \)
19 \( 1 + (0.396 + 0.918i)T \)
23 \( 1 + (0.533 + 0.845i)T \)
29 \( 1 + (-0.360 - 0.932i)T \)
31 \( 1 + (0.952 + 0.305i)T \)
37 \( 1 + (0.973 + 0.230i)T \)
41 \( 1 + (-0.981 - 0.192i)T \)
43 \( 1 + (-0.431 + 0.902i)T \)
47 \( 1 + (0.952 - 0.305i)T \)
53 \( 1 + (-0.939 + 0.342i)T \)
59 \( 1 + (-0.910 - 0.413i)T \)
61 \( 1 + (-0.790 + 0.612i)T \)
67 \( 1 + (-0.360 + 0.932i)T \)
71 \( 1 + (0.893 - 0.448i)T \)
73 \( 1 + (-0.0581 - 0.998i)T \)
79 \( 1 + (0.657 + 0.753i)T \)
83 \( 1 + (-0.981 + 0.192i)T \)
89 \( 1 + (0.893 + 0.448i)T \)
97 \( 1 + (-0.963 - 0.268i)T \)
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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

−26.23599714185204589034623737704, −24.76680601552049501799277627744, −24.30600838085975976634829877783, −23.535501305238331540470450845262, −22.54765032566502435795426161852, −21.8609225991742304620635837283, −20.39617952347275267412079263926, −19.718748062440735380274109566352, −18.48341156323692414858428264211, −17.25269784011825686173362396116, −16.77691506689336753556844719343, −15.66130858250425281407965712901, −14.86573754135663267306496853087, −13.93031676968059890510450857738, −12.95120103603230856229894468334, −11.86432034677353043543256357738, −10.86481201069145162709124987862, −9.25065164442894002147634761371, −8.33208952698084448830928745976, −7.370613621714283986061931521379, −6.619099242113644624999301494358, −4.96406907208925556929331290661, −4.33225066468846472443355011083, −3.154858731547700899971687630913, −0.63368527847537640722771103145, 1.6448445061906518208249587209, 2.82243391391160020380844997966, 4.10977365468973417264118550536, 4.902194840267565598865854449386, 6.35295063518990713874136948822, 7.77495198411172796344626988189, 8.93956007502139923662662473812, 9.83481357406768858741836046644, 11.22820378007816271504493942028, 11.82822302400743127700712033888, 12.461982586419005989851615115677, 13.87785665842878432724871527653, 14.91412094070638306389369990487, 15.37265500556660475397725531370, 17.044136516737186811865938281764, 18.10201378833134629403264294361, 19.031182789484204294205157327992, 19.67091977301416429410061291922, 20.58320818546454892658698575424, 21.70880677317687635530032168070, 22.365270542871192101305589928610, 23.21052229207558316798550501238, 24.22482240617277457154425304874, 25.03904065871904968401849156964, 26.731890411520250771173884347610

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