Properties

Degree 2
Conductor $ 2 \cdot 3^{3} $
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

\( d \)  =  \(2\)
\( N \)  =  \(54\)    =    \(2 \cdot 3^{3}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{54} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 54,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.7015748253$
$L(\frac12)$  $\approx$  $0.7015748253$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
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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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−19.47812896108741, −18.28463154347558, −17.63372907053849, −16.67449023398863, −15.58696880941000, −14.21020099102705, −13.18302012248398, −11.91184860007480, −10.11526477822370, −9.872045258839872, −8.268048909391662, −6.738609006745370, −5.373630155375055, −2.448509783332633, 2.448509783332633, 5.373630155375055, 6.738609006745370, 8.268048909391662, 9.872045258839872, 10.11526477822370, 11.91184860007480, 13.18302012248398, 14.21020099102705, 15.58696880941000, 16.67449023398863, 17.63372907053849, 18.28463154347558, 19.47812896108741

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