# Properties

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

# Origins

## Dirichlet series

 L(s)  = 1 − 9·5-s − 7-s − 63·11-s − 28·13-s − 72·17-s + 98·19-s − 126·23-s − 44·25-s + 126·29-s − 259·31-s + 9·35-s + 386·37-s + 450·41-s − 34·43-s + 54·47-s − 342·49-s + 693·53-s + 567·55-s − 180·59-s − 280·61-s + 252·65-s − 586·67-s − 504·71-s + 161·73-s + 63·77-s + 440·79-s − 999·83-s + ⋯
 L(s)  = 1 − 0.804·5-s − 0.0539·7-s − 1.72·11-s − 0.597·13-s − 1.02·17-s + 1.18·19-s − 1.14·23-s − 0.351·25-s + 0.806·29-s − 1.50·31-s + 0.0434·35-s + 1.71·37-s + 1.71·41-s − 0.120·43-s + 0.167·47-s − 0.997·49-s + 1.79·53-s + 1.39·55-s − 0.397·59-s − 0.587·61-s + 0.480·65-s − 1.06·67-s − 0.842·71-s + 0.258·73-s + 0.0932·77-s + 0.626·79-s − 1.32·83-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$108$$    =    $$2^{2} \cdot 3^{3}$$ $$\varepsilon$$ = $-1$ motivic weight = $$3$$ character : $\chi_{108} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = $$1$$ Selberg data = $$(2,\ 108,\ (\ :3/2),\ -1)$$ $$L(2)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{5}{2})$$ 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 + 9 T + p^{3} T^{2}$$
7 $$1 + T + p^{3} T^{2}$$
11 $$1 + 63 T + p^{3} T^{2}$$
13 $$1 + 28 T + p^{3} T^{2}$$
17 $$1 + 72 T + p^{3} T^{2}$$
19 $$1 - 98 T + p^{3} T^{2}$$
23 $$1 + 126 T + p^{3} T^{2}$$
29 $$1 - 126 T + p^{3} T^{2}$$
31 $$1 + 259 T + p^{3} T^{2}$$
37 $$1 - 386 T + p^{3} T^{2}$$
41 $$1 - 450 T + p^{3} T^{2}$$
43 $$1 + 34 T + p^{3} T^{2}$$
47 $$1 - 54 T + p^{3} T^{2}$$
53 $$1 - 693 T + p^{3} T^{2}$$
59 $$1 + 180 T + p^{3} T^{2}$$
61 $$1 + 280 T + p^{3} T^{2}$$
67 $$1 + 586 T + p^{3} T^{2}$$
71 $$1 + 504 T + p^{3} T^{2}$$
73 $$1 - 161 T + p^{3} T^{2}$$
79 $$1 - 440 T + p^{3} T^{2}$$
83 $$1 + 999 T + p^{3} T^{2}$$
89 $$1 + 882 T + p^{3} T^{2}$$
97 $$1 + 721 T + p^{3} T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}