# Properties

 Degree 2 Conductor $2^{2} \cdot 3^{3}$ Sign $0.287 + 0.957i$ Motivic weight 5 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.821 + 5.59i)2-s + (−30.6 + 9.19i)4-s + 8.15i·5-s + 63.0i·7-s + (−76.6 − 164. i)8-s + (−45.6 + 6.69i)10-s − 442.·11-s − 76.3·13-s + (−352. + 51.7i)14-s + (855. − 563. i)16-s − 643. i·17-s − 2.18e3i·19-s + (−74.9 − 250. i)20-s + (−363. − 2.47e3i)22-s − 2.73e3·23-s + ⋯
 L(s)  = 1 + (0.145 + 0.989i)2-s + (−0.957 + 0.287i)4-s + 0.145i·5-s + 0.485i·7-s + (−0.423 − 0.906i)8-s + (−0.144 + 0.0211i)10-s − 1.10·11-s − 0.125·13-s + (−0.480 + 0.0705i)14-s + (0.835 − 0.550i)16-s − 0.540i·17-s − 1.39i·19-s + (−0.0419 − 0.139i)20-s + (−0.160 − 1.09i)22-s − 1.07·23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.287 + 0.957i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$108$$    =    $$2^{2} \cdot 3^{3}$$ $$\varepsilon$$ = $0.287 + 0.957i$ motivic weight = $$5$$ character : $\chi_{108} (107, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 108,\ (\ :5/2),\ 0.287 + 0.957i)$$ $$L(3)$$ $$\approx$$ $$0.291469 - 0.216891i$$ $$L(\frac12)$$ $$\approx$$ $$0.291469 - 0.216891i$$ $$L(\frac{7}{2})$$ 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 + (-0.821 - 5.59i)T$$
3 $$1$$
good5 $$1 - 8.15iT - 3.12e3T^{2}$$
7 $$1 - 63.0iT - 1.68e4T^{2}$$
11 $$1 + 442.T + 1.61e5T^{2}$$
13 $$1 + 76.3T + 3.71e5T^{2}$$
17 $$1 + 643. iT - 1.41e6T^{2}$$
19 $$1 + 2.18e3iT - 2.47e6T^{2}$$
23 $$1 + 2.73e3T + 6.43e6T^{2}$$
29 $$1 + 1.50e3iT - 2.05e7T^{2}$$
31 $$1 + 5.50e3iT - 2.86e7T^{2}$$
37 $$1 + 4.82e3T + 6.93e7T^{2}$$
41 $$1 - 1.09e4iT - 1.15e8T^{2}$$
43 $$1 + 8.35e3iT - 1.47e8T^{2}$$
47 $$1 + 1.39e4T + 2.29e8T^{2}$$
53 $$1 + 2.28e4iT - 4.18e8T^{2}$$
59 $$1 + 4.83e4T + 7.14e8T^{2}$$
61 $$1 - 5.10e3T + 8.44e8T^{2}$$
67 $$1 - 3.73e4iT - 1.35e9T^{2}$$
71 $$1 + 7.50e4T + 1.80e9T^{2}$$
73 $$1 + 6.52e4T + 2.07e9T^{2}$$
79 $$1 - 7.32e4iT - 3.07e9T^{2}$$
83 $$1 + 6.52e4T + 3.93e9T^{2}$$
89 $$1 + 9.20e3iT - 5.58e9T^{2}$$
97 $$1 - 1.25e5T + 8.58e9T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}