# Properties

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

# Origins

## Dirichlet series

 L(s)  = 1 − 25·7-s − 427·13-s − 1.71e3·19-s − 3.12e3·25-s − 1.03e4·31-s − 6.66e3·37-s − 3.35e3·43-s − 1.61e4·49-s + 5.69e4·61-s − 3.79e4·67-s + 7.95e4·73-s + 9.08e4·79-s + 1.06e4·91-s + 1.77e5·97-s − 2.11e5·103-s + 1.14e5·109-s + ⋯
 L(s)  = 1 − 0.192·7-s − 0.700·13-s − 1.08·19-s − 25-s − 1.92·31-s − 0.799·37-s − 0.276·43-s − 0.962·49-s + 1.95·61-s − 1.03·67-s + 1.74·73-s + 1.63·79-s + 0.135·91-s + 1.91·97-s − 1.96·103-s + 0.922·109-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 + p^{5} T^{2}$$
7 $$1 + 25 T + p^{5} T^{2}$$
11 $$1 + p^{5} T^{2}$$
13 $$1 + 427 T + p^{5} T^{2}$$
17 $$1 + p^{5} T^{2}$$
19 $$1 + 1711 T + p^{5} T^{2}$$
23 $$1 + p^{5} T^{2}$$
29 $$1 + p^{5} T^{2}$$
31 $$1 + 10324 T + p^{5} T^{2}$$
37 $$1 + 6661 T + p^{5} T^{2}$$
41 $$1 + p^{5} T^{2}$$
43 $$1 + 3352 T + p^{5} T^{2}$$
47 $$1 + p^{5} T^{2}$$
53 $$1 + p^{5} T^{2}$$
59 $$1 + p^{5} T^{2}$$
61 $$1 - 56927 T + p^{5} T^{2}$$
67 $$1 + 37939 T + p^{5} T^{2}$$
71 $$1 + p^{5} T^{2}$$
73 $$1 - 79577 T + p^{5} T^{2}$$
79 $$1 - 90857 T + p^{5} T^{2}$$
83 $$1 + p^{5} T^{2}$$
89 $$1 + p^{5} T^{2}$$
97 $$1 - 177725 T + p^{5} T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}