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Properties

Label 2-3e5-9.5-c0-0-0
Degree $2$
Conductor $243$
Sign $0.766 - 0.642i$
Analytic cond. $0.121272$
Root an. cond. $0.348242$
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  + (−0.5 + 0.866i)4-s + (0.5 + 0.866i)7-s + (0.5 − 0.866i)13-s + (−0.499 − 0.866i)16-s − 19-s + (−0.5 − 0.866i)25-s − 0.999·28-s + (0.5 − 0.866i)31-s − 37-s + (0.5 + 0.866i)43-s + (0.499 + 0.866i)52-s + (−1 − 1.73i)61-s + 0.999·64-s + (−1 + 1.73i)67-s + 2·73-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)4-s + (0.5 + 0.866i)7-s + (0.5 − 0.866i)13-s + (−0.499 − 0.866i)16-s − 19-s + (−0.5 − 0.866i)25-s − 0.999·28-s + (0.5 − 0.866i)31-s − 37-s + (0.5 + 0.866i)43-s + (0.499 + 0.866i)52-s + (−1 − 1.73i)61-s + 0.999·64-s + (−1 + 1.73i)67-s + 2·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(243\)    =    \(3^{5}\)
Sign: $0.766 - 0.642i$
Analytic conductor: \(0.121272\)
Root analytic conductor: \(0.348242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{243} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 243,\ (\ :0),\ 0.766 - 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6914931654\)
\(L(\frac12)\) \(\approx\) \(0.6914931654\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 2T + T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.44649465657986661519678677467, −11.68700520546869352886476834118, −10.60593482334328620591933003771, −9.359735561084049327600639408363, −8.399681262209134790421724523789, −7.88790291501342244301764000697, −6.35025518579984016383557468966, −5.13768281647080276610406630081, −3.93242911116486751096784296401, −2.51848137384896730266859745436, 1.62018576048424919760607377832, 3.92544362162399009671072346811, 4.85908750874754404537864165760, 6.12382593642333497846853950934, 7.17565626156071465312996835555, 8.498228422287741963018967232342, 9.339691471161736146011932408401, 10.49979628243254335606059266199, 10.97270749706753325259547122656, 12.20333845879659080516736824876

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