Properties

Degree 2
Conductor $ 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  − 2·4-s − 7-s + 5·13-s + 4·16-s − 7·19-s − 5·25-s + 2·28-s − 4·31-s + 11·37-s + 8·43-s − 6·49-s − 10·52-s − 61-s − 8·64-s + 5·67-s − 7·73-s + 14·76-s + 17·79-s − 5·91-s − 19·97-s + 10·100-s + 101-s + 103-s + 107-s + 109-s − 4·112-s + 113-s + ⋯
L(s)  = 1  − 4-s − 0.377·7-s + 1.38·13-s + 16-s − 1.60·19-s − 25-s + 0.377·28-s − 0.718·31-s + 1.80·37-s + 1.21·43-s − 6/7·49-s − 1.38·52-s − 0.128·61-s − 64-s + 0.610·67-s − 0.819·73-s + 1.60·76-s + 1.91·79-s − 0.524·91-s − 1.92·97-s + 100-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s − 0.377·112-s + 0.0940·113-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(27\)    =    \(3^{3}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{27} (1, \cdot )$
Sato-Tate  :  $N(\mathrm{U}(1))$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 27,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.5888795834$
$L(\frac12)$  $\approx$  $0.5888795834$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 3$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p = 3$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 \)
good2 \( 1 + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 - 17 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 19 T + p T^{2} \)
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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

−19.50763388930887, −18.55924110538682, −17.54958210256512, −16.30001713525188, −14.89211076740925, −13.58211369833952, −12.71563949069013, −10.90872829298909, −9.429199208210394, −8.217650367462527, −6.048935400009874, −4.043044013797433, 4.043044013797433, 6.048935400009874, 8.217650367462527, 9.429199208210394, 10.90872829298909, 12.71563949069013, 13.58211369833952, 14.89211076740925, 16.30001713525188, 17.54958210256512, 18.55924110538682, 19.50763388930887

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