Properties

Degree 2
Conductor $ 3^{3} $
Sign $-1$
Motivic weight 3
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s + 4-s − 15·5-s − 25·7-s + 21·8-s + 45·10-s + 15·11-s + 20·13-s + 75·14-s − 71·16-s − 72·17-s + 2·19-s − 15·20-s − 45·22-s − 114·23-s + 100·25-s − 60·26-s − 25·28-s − 30·29-s + 101·31-s + 45·32-s + 216·34-s + 375·35-s − 430·37-s − 6·38-s − 315·40-s + 30·41-s + ⋯
L(s)  = 1  − 1.06·2-s + 1/8·4-s − 1.34·5-s − 1.34·7-s + 0.928·8-s + 1.42·10-s + 0.411·11-s + 0.426·13-s + 1.43·14-s − 1.10·16-s − 1.02·17-s + 0.0241·19-s − 0.167·20-s − 0.436·22-s − 1.03·23-s + 4/5·25-s − 0.452·26-s − 0.168·28-s − 0.192·29-s + 0.585·31-s + 0.248·32-s + 1.08·34-s + 1.81·35-s − 1.91·37-s − 0.0256·38-s − 1.24·40-s + 0.114·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(27\)    =    \(3^{3}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(3\)
character  :  $\chi_{27} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 27,\ (\ :3/2),\ -1)$
$L(2)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{5}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 3$,\(F_p(T)\) is a polynomial of degree 2. If $p = 3$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 + 3 T + p^{3} T^{2} \)
5 \( 1 + 3 p T + p^{3} T^{2} \)
7 \( 1 + 25 T + p^{3} T^{2} \)
11 \( 1 - 15 T + p^{3} T^{2} \)
13 \( 1 - 20 T + p^{3} T^{2} \)
17 \( 1 + 72 T + p^{3} T^{2} \)
19 \( 1 - 2 T + p^{3} T^{2} \)
23 \( 1 + 114 T + p^{3} T^{2} \)
29 \( 1 + 30 T + p^{3} T^{2} \)
31 \( 1 - 101 T + p^{3} T^{2} \)
37 \( 1 + 430 T + p^{3} T^{2} \)
41 \( 1 - 30 T + p^{3} T^{2} \)
43 \( 1 - 110 T + p^{3} T^{2} \)
47 \( 1 - 330 T + p^{3} T^{2} \)
53 \( 1 + 621 T + p^{3} T^{2} \)
59 \( 1 - 660 T + p^{3} T^{2} \)
61 \( 1 + 376 T + p^{3} T^{2} \)
67 \( 1 + 250 T + p^{3} T^{2} \)
71 \( 1 - 360 T + p^{3} T^{2} \)
73 \( 1 - 785 T + p^{3} T^{2} \)
79 \( 1 - 488 T + p^{3} T^{2} \)
83 \( 1 + 489 T + p^{3} T^{2} \)
89 \( 1 - 450 T + p^{3} T^{2} \)
97 \( 1 + 1105 T + p^{3} T^{2} \)
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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

−16.20916996209474387110605450378, −15.61931700275980054748424927049, −13.66288744850342102249283282678, −12.24042256095807950623660927833, −10.80893294264929195054135441410, −9.422115507214629431427816192925, −8.248266150907710843417869845466, −6.82607085022633208400724953775, −3.94331193213095727698319461419, 0, 3.94331193213095727698319461419, 6.82607085022633208400724953775, 8.248266150907710843417869845466, 9.422115507214629431427816192925, 10.80893294264929195054135441410, 12.24042256095807950623660927833, 13.66288744850342102249283282678, 15.61931700275980054748424927049, 16.20916996209474387110605450378

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