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Properties

Label 1-3e5-243.151-r0-0-0
Degree $1$
Conductor $243$
Sign $0.527 - 0.849i$
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  + (0.856 − 0.516i)2-s + (0.466 − 0.884i)4-s + (0.996 + 0.0774i)5-s + (0.952 − 0.305i)7-s + (−0.0581 − 0.998i)8-s + (0.893 − 0.448i)10-s + (−0.963 − 0.268i)11-s + (−0.360 + 0.932i)13-s + (0.657 − 0.753i)14-s + (−0.565 − 0.824i)16-s + (0.597 + 0.802i)17-s + (−0.993 − 0.116i)19-s + (0.533 − 0.845i)20-s + (−0.963 + 0.268i)22-s + (−0.740 + 0.672i)23-s + ⋯
L(s)  = 1  + (0.856 − 0.516i)2-s + (0.466 − 0.884i)4-s + (0.996 + 0.0774i)5-s + (0.952 − 0.305i)7-s + (−0.0581 − 0.998i)8-s + (0.893 − 0.448i)10-s + (−0.963 − 0.268i)11-s + (−0.360 + 0.932i)13-s + (0.657 − 0.753i)14-s + (−0.565 − 0.824i)16-s + (0.597 + 0.802i)17-s + (−0.993 − 0.116i)19-s + (0.533 − 0.845i)20-s + (−0.963 + 0.268i)22-s + (−0.740 + 0.672i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.527 - 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.527 - 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(243\)    =    \(3^{5}\)
Sign: $0.527 - 0.849i$
Analytic conductor: \(1.12848\)
Root analytic conductor: \(1.12848\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{243} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 243,\ (0:\ ),\ 0.527 - 0.849i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.017176716 - 1.121550931i\)
\(L(\frac12)\) \(\approx\) \(2.017176716 - 1.121550931i\)
\(L(1)\) \(\approx\) \(1.783976941 - 0.6698228757i\)
\(L(1)\) \(\approx\) \(1.783976941 - 0.6698228757i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 + (0.856 - 0.516i)T \)
5 \( 1 + (0.996 + 0.0774i)T \)
7 \( 1 + (0.952 - 0.305i)T \)
11 \( 1 + (-0.963 - 0.268i)T \)
13 \( 1 + (-0.360 + 0.932i)T \)
17 \( 1 + (0.597 + 0.802i)T \)
19 \( 1 + (-0.993 - 0.116i)T \)
23 \( 1 + (-0.740 + 0.672i)T \)
29 \( 1 + (0.323 - 0.946i)T \)
31 \( 1 + (-0.135 + 0.990i)T \)
37 \( 1 + (-0.286 - 0.957i)T \)
41 \( 1 + (-0.875 + 0.483i)T \)
43 \( 1 + (0.0968 - 0.995i)T \)
47 \( 1 + (-0.135 - 0.990i)T \)
53 \( 1 + (-0.939 + 0.342i)T \)
59 \( 1 + (0.713 + 0.700i)T \)
61 \( 1 + (0.466 + 0.884i)T \)
67 \( 1 + (0.323 + 0.946i)T \)
71 \( 1 + (-0.835 - 0.549i)T \)
73 \( 1 + (0.893 + 0.448i)T \)
79 \( 1 + (0.0193 + 0.999i)T \)
83 \( 1 + (-0.875 - 0.483i)T \)
89 \( 1 + (-0.835 + 0.549i)T \)
97 \( 1 + (0.996 - 0.0774i)T \)
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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

−25.83668134893986355704736826677, −25.32901066837037389931707689699, −24.40930906464261096459189589432, −23.68167717037592782061095302289, −22.54069189217088299819203556983, −21.77144539833682144617767262537, −20.79823385883127066563300357846, −20.46205357530045299676812080194, −18.490654347081669801324730485183, −17.73010073796557506460134241497, −16.930304802041470481917054195769, −15.78394585339375997652233948594, −14.780211963352715339405432478242, −14.12916005234981319687055555483, −13.02869417449897534046395787723, −12.34393668401480671110121269500, −11.02736459730264981515248127310, −9.99758493740434716071853801613, −8.45852431577741752920640614409, −7.68705894949010119840654011184, −6.34061797134721284924478875476, −5.30285506602821614027023817690, −4.74346312463835427168340334082, −2.93352875885656413209763534646, −1.99618290817274817430391819467, 1.59804701506074003285780275466, 2.39256279969112152898592676190, 3.9471811583429907592512050581, 5.067985781439023321017219224298, 5.89363725154935260753947877327, 7.0969131934220173141742895826, 8.54258576714092198142028245424, 10.00446839994120393090185064023, 10.605506728186070580056188996663, 11.675026356328438156312170866356, 12.77784464334425310040489492287, 13.74623370611872124168715780766, 14.33362685251035210298680158504, 15.30360148897998437124378615247, 16.629822442189472266621276937067, 17.648693012610550524042570159567, 18.66988858276784103202195694886, 19.63642258341317840761789745958, 20.95516276448791158976626503799, 21.28282037577616975256824825978, 21.97254846008435482876058949261, 23.48509308609163718994768470233, 23.82011806079307593131310368643, 24.8903339024106172216312383770, 25.86082717228271725807792426291

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