Properties

Degree $2$
Conductor $5184$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·5-s − 4·7-s + 13-s − 3·17-s + 4·19-s + 4·25-s − 9·29-s − 4·31-s − 12·35-s + 37-s + 6·41-s − 8·43-s − 12·47-s + 9·49-s + 6·53-s + 61-s + 3·65-s + 4·67-s − 12·71-s + 11·73-s − 16·79-s + 12·83-s − 9·85-s − 3·89-s − 4·91-s + 12·95-s + 2·97-s + ⋯
L(s)  = 1  + 1.34·5-s − 1.51·7-s + 0.277·13-s − 0.727·17-s + 0.917·19-s + 4/5·25-s − 1.67·29-s − 0.718·31-s − 2.02·35-s + 0.164·37-s + 0.937·41-s − 1.21·43-s − 1.75·47-s + 9/7·49-s + 0.824·53-s + 0.128·61-s + 0.372·65-s + 0.488·67-s − 1.42·71-s + 1.28·73-s − 1.80·79-s + 1.31·83-s − 0.976·85-s − 0.317·89-s − 0.419·91-s + 1.23·95-s + 0.203·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5184\)    =    \(2^{6} \cdot 3^{4}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{5184} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5184,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 - 3 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 3 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−7.77133374356300284680128512104, −6.89070553337933054938428171633, −6.42718799904595728230962453845, −5.72247365022694368090686428400, −5.20981735085530076681492595480, −3.96148093445634060055559370451, −3.22684654073668095375893389695, −2.39288338822089294054098347965, −1.47843412021064514807145846869, 0, 1.47843412021064514807145846869, 2.39288338822089294054098347965, 3.22684654073668095375893389695, 3.96148093445634060055559370451, 5.20981735085530076681492595480, 5.72247365022694368090686428400, 6.42718799904595728230962453845, 6.89070553337933054938428171633, 7.77133374356300284680128512104

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