# Properties

 Degree 2 Conductor $2^{2} \cdot 3^{3}$ Sign $i$ Motivic weight 4 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 9i·5-s + 5·7-s − 117i·11-s − 34·13-s − 450i·17-s − 64·19-s − 612i·23-s + 544·25-s − 1.06e3i·29-s − 697·31-s − 45i·35-s − 748·37-s − 684i·41-s + 2.61e3·43-s + 2.64e3i·47-s + ⋯
 L(s)  = 1 − 0.359i·5-s + 0.102·7-s − 0.966i·11-s − 0.201·13-s − 1.55i·17-s − 0.177·19-s − 1.15i·23-s + 0.870·25-s − 1.26i·29-s − 0.725·31-s − 0.0367i·35-s − 0.546·37-s − 0.406i·41-s + 1.41·43-s + 1.19i·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$108$$    =    $$2^{2} \cdot 3^{3}$$ $$\varepsilon$$ = $i$ motivic weight = $$4$$ character : $\chi_{108} (53, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 108,\ (\ :2),\ i)$$ $$L(\frac{5}{2})$$ $$\approx$$ $$1.00366 - 1.00366i$$ $$L(\frac12)$$ $$\approx$$ $$1.00366 - 1.00366i$$ $$L(3)$$ 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 + 9iT - 625T^{2}$$
7 $$1 - 5T + 2.40e3T^{2}$$
11 $$1 + 117iT - 1.46e4T^{2}$$
13 $$1 + 34T + 2.85e4T^{2}$$
17 $$1 + 450iT - 8.35e4T^{2}$$
19 $$1 + 64T + 1.30e5T^{2}$$
23 $$1 + 612iT - 2.79e5T^{2}$$
29 $$1 + 1.06e3iT - 7.07e5T^{2}$$
31 $$1 + 697T + 9.23e5T^{2}$$
37 $$1 + 748T + 1.87e6T^{2}$$
41 $$1 + 684iT - 2.82e6T^{2}$$
43 $$1 - 2.61e3T + 3.41e6T^{2}$$
47 $$1 - 2.64e3iT - 4.87e6T^{2}$$
53 $$1 - 1.07e3iT - 7.89e6T^{2}$$
59 $$1 - 5.81e3iT - 1.21e7T^{2}$$
61 $$1 - 6.40e3T + 1.38e7T^{2}$$
67 $$1 + 5.21e3T + 2.01e7T^{2}$$
71 $$1 - 6.57e3iT - 2.54e7T^{2}$$
73 $$1 + 4.51e3T + 2.83e7T^{2}$$
79 $$1 - 7.50e3T + 3.89e7T^{2}$$
83 $$1 - 5.48e3iT - 4.74e7T^{2}$$
89 $$1 + 8.87e3iT - 6.27e7T^{2}$$
97 $$1 - 1.05e4T + 8.85e7T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}