Properties

Label 2-3e3-3.2-c2-0-1
Degree $2$
Conductor $27$
Sign $1$
Analytic cond. $0.735696$
Root an. cond. $0.857727$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·4-s − 13·7-s − 13-s + 16·16-s + 11·19-s + 25·25-s − 52·28-s − 46·31-s + 47·37-s − 22·43-s + 120·49-s − 4·52-s − 121·61-s + 64·64-s − 109·67-s − 97·73-s + 44·76-s + 131·79-s + 13·91-s + 167·97-s + 100·100-s − 37·103-s − 214·109-s − 208·112-s + ⋯
L(s)  = 1  + 4-s − 1.85·7-s − 0.0769·13-s + 16-s + 0.578·19-s + 25-s − 1.85·28-s − 1.48·31-s + 1.27·37-s − 0.511·43-s + 2.44·49-s − 0.0769·52-s − 1.98·61-s + 64-s − 1.62·67-s − 1.32·73-s + 0.578·76-s + 1.65·79-s + 1/7·91-s + 1.72·97-s + 100-s − 0.359·103-s − 1.96·109-s − 1.85·112-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(27\)    =    \(3^{3}\)
Sign: $1$
Analytic conductor: \(0.735696\)
Root analytic conductor: \(0.857727\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: $\chi_{27} (26, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 27,\ (\ :1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.000965941\)
\(L(\frac12)\) \(\approx\) \(1.000965941\)
\(L(2)\) 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 \( ( 1 - p T )( 1 + p T ) \)
5 \( ( 1 - p T )( 1 + p T ) \)
7 \( 1 + 13 T + p^{2} T^{2} \)
11 \( ( 1 - p T )( 1 + p T ) \)
13 \( 1 + T + p^{2} T^{2} \)
17 \( ( 1 - p T )( 1 + p T ) \)
19 \( 1 - 11 T + p^{2} T^{2} \)
23 \( ( 1 - p T )( 1 + p T ) \)
29 \( ( 1 - p T )( 1 + p T ) \)
31 \( 1 + 46 T + p^{2} T^{2} \)
37 \( 1 - 47 T + p^{2} T^{2} \)
41 \( ( 1 - p T )( 1 + p T ) \)
43 \( 1 + 22 T + p^{2} T^{2} \)
47 \( ( 1 - p T )( 1 + p T ) \)
53 \( ( 1 - p T )( 1 + p T ) \)
59 \( ( 1 - p T )( 1 + p T ) \)
61 \( 1 + 121 T + p^{2} T^{2} \)
67 \( 1 + 109 T + p^{2} T^{2} \)
71 \( ( 1 - p T )( 1 + p T ) \)
73 \( 1 + 97 T + p^{2} T^{2} \)
79 \( 1 - 131 T + p^{2} T^{2} \)
83 \( ( 1 - p T )( 1 + p T ) \)
89 \( ( 1 - p T )( 1 + p T ) \)
97 \( 1 - 167 T + p^{2} T^{2} \)
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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

−16.66700335745736228791898170610, −16.12719347488778567737906039955, −14.93317694987577442968540511904, −13.18826623214618288315647120749, −12.16309481374790876133361323529, −10.64354618326625880843609971954, −9.385096739194486683174830096085, −7.25050227093460265001191315374, −6.08002078378355951356788956969, −3.12785755550327188554888192988, 3.12785755550327188554888192988, 6.08002078378355951356788956969, 7.25050227093460265001191315374, 9.385096739194486683174830096085, 10.64354618326625880843609971954, 12.16309481374790876133361323529, 13.18826623214618288315647120749, 14.93317694987577442968540511904, 16.12719347488778567737906039955, 16.66700335745736228791898170610

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