Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $1$
Motivic weight 5
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 88.1·5-s + 29·7-s + 88.1·11-s + 329·13-s − 2.20e3·17-s + 1.79e3·19-s + 3.61e3·23-s + 4.65e3·25-s + 1.41e3·29-s + 5.22e3·31-s + 2.55e3·35-s + 8.78e3·37-s − 1.55e4·41-s + 1.99e4·43-s − 1.08e4·47-s − 1.59e4·49-s + 2.94e4·53-s + 7.77e3·55-s + 5.73e3·59-s − 1.06e3·61-s + 2.90e4·65-s − 6.20e4·67-s − 4.63e4·71-s − 4.80e4·73-s + 2.55e3·77-s + 4.99e4·79-s − 5.76e4·83-s + ⋯
L(s)  = 1  + 1.57·5-s + 0.223·7-s + 0.219·11-s + 0.539·13-s − 1.85·17-s + 1.14·19-s + 1.42·23-s + 1.48·25-s + 0.311·29-s + 0.977·31-s + 0.352·35-s + 1.05·37-s − 1.44·41-s + 1.64·43-s − 0.716·47-s − 0.949·49-s + 1.44·53-s + 0.346·55-s + 0.214·59-s − 0.0367·61-s + 0.851·65-s − 1.68·67-s − 1.09·71-s − 1.05·73-s + 0.0491·77-s + 0.900·79-s − 0.918·83-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  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :5/2),\ 1)\)
\(L(3)\)  \(\approx\)  \(2.58575\)
\(L(\frac12)\)  \(\approx\)  \(2.58575\)
\(L(\frac{7}{2})\)   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 \( 1 - 88.1T + 3.12e3T^{2} \)
7 \( 1 - 29T + 1.68e4T^{2} \)
11 \( 1 - 88.1T + 1.61e5T^{2} \)
13 \( 1 - 329T + 3.71e5T^{2} \)
17 \( 1 + 2.20e3T + 1.41e6T^{2} \)
19 \( 1 - 1.79e3T + 2.47e6T^{2} \)
23 \( 1 - 3.61e3T + 6.43e6T^{2} \)
29 \( 1 - 1.41e3T + 2.05e7T^{2} \)
31 \( 1 - 5.22e3T + 2.86e7T^{2} \)
37 \( 1 - 8.78e3T + 6.93e7T^{2} \)
41 \( 1 + 1.55e4T + 1.15e8T^{2} \)
43 \( 1 - 1.99e4T + 1.47e8T^{2} \)
47 \( 1 + 1.08e4T + 2.29e8T^{2} \)
53 \( 1 - 2.94e4T + 4.18e8T^{2} \)
59 \( 1 - 5.73e3T + 7.14e8T^{2} \)
61 \( 1 + 1.06e3T + 8.44e8T^{2} \)
67 \( 1 + 6.20e4T + 1.35e9T^{2} \)
71 \( 1 + 4.63e4T + 1.80e9T^{2} \)
73 \( 1 + 4.80e4T + 2.07e9T^{2} \)
79 \( 1 - 4.99e4T + 3.07e9T^{2} \)
83 \( 1 + 5.76e4T + 3.93e9T^{2} \)
89 \( 1 + 8.79e4T + 5.58e9T^{2} \)
97 \( 1 - 1.29e4T + 8.58e9T^{2} \)
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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

−13.19511601443184025649020698767, −11.59436403364964920292591836039, −10.58233259051719470918443246567, −9.483382057989998486003113401563, −8.689545064228936692991465487764, −6.93499565280773669993965702308, −5.94610751945138490172786722389, −4.71498481783240816581428689664, −2.69793736432408052759390939110, −1.30593061073762158092596882428, 1.30593061073762158092596882428, 2.69793736432408052759390939110, 4.71498481783240816581428689664, 5.94610751945138490172786722389, 6.93499565280773669993965702308, 8.689545064228936692991465487764, 9.483382057989998486003113401563, 10.58233259051719470918443246567, 11.59436403364964920292591836039, 13.19511601443184025649020698767

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