Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-1$
Motivic weight 5
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 57.6·5-s + 156.·7-s + 218.·11-s − 587.·13-s − 2.12e3·17-s + 29.7·19-s − 3.17e3·23-s + 196·25-s − 3.64e3·29-s + 6.33e3·31-s − 9.04e3·35-s − 8.43e3·37-s − 1.01e4·41-s − 6.85e3·43-s − 2.44e4·47-s + 7.80e3·49-s + 2.54e4·53-s − 1.25e4·55-s + 2.06e4·59-s + 4.55e4·61-s + 3.38e4·65-s + 1.90e4·67-s − 2.30e4·71-s − 8.96e4·73-s + 3.42e4·77-s − 3.40e4·79-s + 1.08e5·83-s + ⋯
L(s)  = 1  − 1.03·5-s + 1.21·7-s + 0.543·11-s − 0.964·13-s − 1.78·17-s + 0.0189·19-s − 1.25·23-s + 0.0627·25-s − 0.804·29-s + 1.18·31-s − 1.24·35-s − 1.01·37-s − 0.942·41-s − 0.565·43-s − 1.61·47-s + 0.464·49-s + 1.24·53-s − 0.560·55-s + 0.774·59-s + 1.56·61-s + 0.994·65-s + 0.519·67-s − 0.542·71-s − 1.96·73-s + 0.657·77-s − 0.613·79-s + 1.72·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(5\)
character  :  $\chi_{108} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 108,\ (\ :5/2),\ -1)\)
\(L(3)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{7}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + 57.6T + 3.12e3T^{2} \)
7 \( 1 - 156.T + 1.68e4T^{2} \)
11 \( 1 - 218.T + 1.61e5T^{2} \)
13 \( 1 + 587.T + 3.71e5T^{2} \)
17 \( 1 + 2.12e3T + 1.41e6T^{2} \)
19 \( 1 - 29.7T + 2.47e6T^{2} \)
23 \( 1 + 3.17e3T + 6.43e6T^{2} \)
29 \( 1 + 3.64e3T + 2.05e7T^{2} \)
31 \( 1 - 6.33e3T + 2.86e7T^{2} \)
37 \( 1 + 8.43e3T + 6.93e7T^{2} \)
41 \( 1 + 1.01e4T + 1.15e8T^{2} \)
43 \( 1 + 6.85e3T + 1.47e8T^{2} \)
47 \( 1 + 2.44e4T + 2.29e8T^{2} \)
53 \( 1 - 2.54e4T + 4.18e8T^{2} \)
59 \( 1 - 2.06e4T + 7.14e8T^{2} \)
61 \( 1 - 4.55e4T + 8.44e8T^{2} \)
67 \( 1 - 1.90e4T + 1.35e9T^{2} \)
71 \( 1 + 2.30e4T + 1.80e9T^{2} \)
73 \( 1 + 8.96e4T + 2.07e9T^{2} \)
79 \( 1 + 3.40e4T + 3.07e9T^{2} \)
83 \( 1 - 1.08e5T + 3.93e9T^{2} \)
89 \( 1 - 1.10e5T + 5.58e9T^{2} \)
97 \( 1 + 2.88e4T + 8.58e9T^{2} \)
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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

−11.73221934650396198378854446496, −11.63273864058172205143549725506, −10.20969250921568759555062763524, −8.719751327649295135810788781785, −7.88475169166221899770692311595, −6.75818820268687459738952615895, −4.94224891656756950196435959752, −3.98570935811126743670208001395, −1.99809270355761564298561986659, 0, 1.99809270355761564298561986659, 3.98570935811126743670208001395, 4.94224891656756950196435959752, 6.75818820268687459738952615895, 7.88475169166221899770692311595, 8.719751327649295135810788781785, 10.20969250921568759555062763524, 11.63273864058172205143549725506, 11.73221934650396198378854446496

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