# Properties

 Degree 2 Conductor $2^{2} \cdot 3^{3}$ Sign $0.633 - 0.773i$ Motivic weight 5 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (1.78 − 5.36i)2-s + (−25.6 − 19.1i)4-s + (−12.8 + 7.42i)5-s + (−15.2 − 8.83i)7-s + (−148. + 103. i)8-s + (16.9 + 82.2i)10-s + (−143. + 248. i)11-s + (42.2 + 73.1i)13-s + (−74.6 + 66.3i)14-s + (291. + 981. i)16-s + 1.60e3i·17-s − 1.66e3i·19-s + (472. + 55.6i)20-s + (1.07e3 + 1.21e3i)22-s + (2.32e3 + 4.02e3i)23-s + ⋯
 L(s)  = 1 + (0.315 − 0.949i)2-s + (−0.801 − 0.598i)4-s + (−0.230 + 0.132i)5-s + (−0.118 − 0.0681i)7-s + (−0.820 + 0.572i)8-s + (0.0535 + 0.260i)10-s + (−0.356 + 0.618i)11-s + (0.0693 + 0.120i)13-s + (−0.101 + 0.0905i)14-s + (0.284 + 0.958i)16-s + 1.35i·17-s − 1.06i·19-s + (0.263 + 0.0311i)20-s + (0.474 + 0.533i)22-s + (0.916 + 1.58i)23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.633 - 0.773i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.633 - 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$108$$    =    $$2^{2} \cdot 3^{3}$$ $$\varepsilon$$ = $0.633 - 0.773i$ motivic weight = $$5$$ character : $\chi_{108} (35, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 108,\ (\ :5/2),\ 0.633 - 0.773i)$$ $$L(3)$$ $$\approx$$ $$0.850665 + 0.402831i$$ $$L(\frac12)$$ $$\approx$$ $$0.850665 + 0.402831i$$ $$L(\frac{7}{2})$$ 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 + (-1.78 + 5.36i)T$$
3 $$1$$
good5 $$1 + (12.8 - 7.42i)T + (1.56e3 - 2.70e3i)T^{2}$$
7 $$1 + (15.2 + 8.83i)T + (8.40e3 + 1.45e4i)T^{2}$$
11 $$1 + (143. - 248. i)T + (-8.05e4 - 1.39e5i)T^{2}$$
13 $$1 + (-42.2 - 73.1i)T + (-1.85e5 + 3.21e5i)T^{2}$$
17 $$1 - 1.60e3iT - 1.41e6T^{2}$$
19 $$1 + 1.66e3iT - 2.47e6T^{2}$$
23 $$1 + (-2.32e3 - 4.02e3i)T + (-3.21e6 + 5.57e6i)T^{2}$$
29 $$1 + (2.41e3 + 1.39e3i)T + (1.02e7 + 1.77e7i)T^{2}$$
31 $$1 + (-2.78e3 + 1.60e3i)T + (1.43e7 - 2.47e7i)T^{2}$$
37 $$1 + 4.62e3T + 6.93e7T^{2}$$
41 $$1 + (9.81e3 - 5.66e3i)T + (5.79e7 - 1.00e8i)T^{2}$$
43 $$1 + (-1.23e4 - 7.15e3i)T + (7.35e7 + 1.27e8i)T^{2}$$
47 $$1 + (-1.69e3 + 2.93e3i)T + (-1.14e8 - 1.98e8i)T^{2}$$
53 $$1 - 3.10e4iT - 4.18e8T^{2}$$
59 $$1 + (1.09e4 + 1.88e4i)T + (-3.57e8 + 6.19e8i)T^{2}$$
61 $$1 + (1.78e4 - 3.09e4i)T + (-4.22e8 - 7.31e8i)T^{2}$$
67 $$1 + (1.80e4 - 1.04e4i)T + (6.75e8 - 1.16e9i)T^{2}$$
71 $$1 + 4.22e4T + 1.80e9T^{2}$$
73 $$1 + 7.29e4T + 2.07e9T^{2}$$
79 $$1 + (7.60e4 + 4.39e4i)T + (1.53e9 + 2.66e9i)T^{2}$$
83 $$1 + (-1.22e4 + 2.11e4i)T + (-1.96e9 - 3.41e9i)T^{2}$$
89 $$1 - 1.19e5iT - 5.58e9T^{2}$$
97 $$1 + (-3.95e4 + 6.85e4i)T + (-4.29e9 - 7.43e9i)T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}