# Properties

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

# Origins

## Dirichlet series

 L(s)  = 1 + 11·7-s + 23·13-s − 37·19-s + 25·25-s − 46·31-s − 73·37-s − 22·43-s + 72·49-s + 47·61-s − 13·67-s + 143·73-s + 11·79-s + 253·91-s − 169·97-s − 157·103-s − 214·109-s + ⋯
 L(s)  = 1 + 11/7·7-s + 1.76·13-s − 1.94·19-s + 25-s − 1.48·31-s − 1.97·37-s − 0.511·43-s + 1.46·49-s + 0.770·61-s − 0.194·67-s + 1.95·73-s + 0.139·79-s + 2.78·91-s − 1.74·97-s − 1.52·103-s − 1.96·109-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$108$$    =    $$2^{2} \cdot 3^{3}$$ $$\varepsilon$$ = $1$ motivic weight = $$2$$ character : $\chi_{108} (53, \cdot )$ primitive : yes self-dual : yes analytic rank = $$0$$ Selberg data = $$(2,\ 108,\ (\ :1),\ 1)$$ $$L(\frac{3}{2})$$ $$\approx$$ $$1.54717$$ $$L(\frac12)$$ $$\approx$$ $$1.54717$$ $$L(2)$$ 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 - p T )( 1 + p T )$$
7 $$1 - 11 T + p^{2} T^{2}$$
11 $$( 1 - p T )( 1 + p T )$$
13 $$1 - 23 T + p^{2} T^{2}$$
17 $$( 1 - p T )( 1 + p T )$$
19 $$1 + 37 T + p^{2} T^{2}$$
23 $$( 1 - p T )( 1 + p T )$$
29 $$( 1 - p T )( 1 + p T )$$
31 $$1 + 46 T + p^{2} T^{2}$$
37 $$1 + 73 T + p^{2} T^{2}$$
41 $$( 1 - p T )( 1 + p T )$$
43 $$1 + 22 T + p^{2} T^{2}$$
47 $$( 1 - p T )( 1 + p T )$$
53 $$( 1 - p T )( 1 + p T )$$
59 $$( 1 - p T )( 1 + p T )$$
61 $$1 - 47 T + p^{2} T^{2}$$
67 $$1 + 13 T + p^{2} T^{2}$$
71 $$( 1 - p T )( 1 + p T )$$
73 $$1 - 143 T + p^{2} T^{2}$$
79 $$1 - 11 T + p^{2} T^{2}$$
83 $$( 1 - p T )( 1 + p T )$$
89 $$( 1 - p T )( 1 + p T )$$
97 $$1 + 169 T + p^{2} T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}