Properties

Degree $2$
Conductor $54$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 3·5-s − 7-s + 8-s − 3·10-s + 3·11-s − 4·13-s − 14-s + 16-s + 2·19-s − 3·20-s + 3·22-s + 6·23-s + 4·25-s − 4·26-s − 28-s − 6·29-s + 5·31-s + 32-s + 3·35-s + 2·37-s + 2·38-s − 3·40-s + 6·41-s − 10·43-s + 3·44-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 1.34·5-s − 0.377·7-s + 0.353·8-s − 0.948·10-s + 0.904·11-s − 1.10·13-s − 0.267·14-s + 1/4·16-s + 0.458·19-s − 0.670·20-s + 0.639·22-s + 1.25·23-s + 4/5·25-s − 0.784·26-s − 0.188·28-s − 1.11·29-s + 0.898·31-s + 0.176·32-s + 0.507·35-s + 0.328·37-s + 0.324·38-s − 0.474·40-s + 0.937·41-s − 1.52·43-s + 0.452·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(54\)    =    \(2 \cdot 3^{3}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{54} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 54,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.030521849\)
\(L(\frac12)\) \(\approx\) \(1.030521849\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
good5 \( 1 + 3 T + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−19.59734618587691, −18.97677914698781, −17.20809020101060, −16.25220226241128, −15.21324045809464, −14.47948142417272, −12.98699917725806, −11.98990591115535, −11.21947207780375, −9.512741326522108, −7.827238430940929, −6.715087259886294, −4.777093331123716, −3.385491783694135, 3.385491783694135, 4.777093331123716, 6.715087259886294, 7.827238430940929, 9.512741326522108, 11.21947207780375, 11.98990591115535, 12.98699917725806, 14.47948142417272, 15.21324045809464, 16.25220226241128, 17.20809020101060, 18.97677914698781, 19.59734618587691

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