# Properties

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

# Origins

## Dirichlet series

 L(s)  = 1 − 7-s − 13-s − 19-s + 25-s + 2·31-s − 37-s + 2·43-s − 61-s − 67-s − 73-s − 79-s + 91-s − 97-s − 103-s + 2·109-s + ⋯
 L(s)  = 1 − 7-s − 13-s − 19-s + 25-s + 2·31-s − 37-s + 2·43-s − 61-s − 67-s − 73-s − 79-s + 91-s − 97-s − 103-s + 2·109-s + ⋯

## Functional equation

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

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$108$$    =    $$2^{2} \cdot 3^{3}$$ $$\varepsilon$$ = $1$ motivic weight = $$0$$ character : $\chi_{108} (53, \cdot )$ primitive : yes self-dual : yes analytic rank = $$0$$ Selberg data = $$(2,\ 108,\ (\ :0),\ 1)$$ $$L(\frac{1}{2})$$ $$\approx$$ $$0.5402022062$$ $$L(\frac12)$$ $$\approx$$ $$0.5402022062$$ $$L(1)$$ 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 - T )( 1 + T )$$
7 $$1 + T + T^{2}$$
11 $$( 1 - T )( 1 + T )$$
13 $$1 + T + T^{2}$$
17 $$( 1 - T )( 1 + T )$$
19 $$1 + T + T^{2}$$
23 $$( 1 - T )( 1 + T )$$
29 $$( 1 - T )( 1 + T )$$
31 $$( 1 - T )^{2}$$
37 $$1 + T + T^{2}$$
41 $$( 1 - T )( 1 + T )$$
43 $$( 1 - T )^{2}$$
47 $$( 1 - T )( 1 + T )$$
53 $$( 1 - T )( 1 + T )$$
59 $$( 1 - T )( 1 + T )$$
61 $$1 + T + T^{2}$$
67 $$1 + T + T^{2}$$
71 $$( 1 - T )( 1 + T )$$
73 $$1 + T + T^{2}$$
79 $$1 + T + T^{2}$$
83 $$( 1 - T )( 1 + T )$$
89 $$( 1 - T )( 1 + T )$$
97 $$1 + T + T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}