# Properties

 Degree 2 Conductor $2^{2} \cdot 3^{3}$ Sign $0.566 - 0.824i$ Motivic weight 3 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.44 − 2.43i)2-s + (−3.82 + 7.02i)4-s + (−14.6 − 8.45i)5-s + (3.08 − 1.78i)7-s + (22.6 − 0.866i)8-s + (0.610 + 47.8i)10-s + (25.0 + 43.4i)11-s + (−18.9 + 32.9i)13-s + (−8.80 − 4.93i)14-s + (−34.7 − 53.7i)16-s + 84.3i·17-s + 62.9i·19-s + (115. − 70.6i)20-s + (69.3 − 123. i)22-s + (37.6 − 65.1i)23-s + ⋯
 L(s)  = 1 + (−0.511 − 0.859i)2-s + (−0.477 + 0.878i)4-s + (−1.31 − 0.756i)5-s + (0.166 − 0.0963i)7-s + (0.999 − 0.0382i)8-s + (0.0193 + 1.51i)10-s + (0.687 + 1.19i)11-s + (−0.405 + 0.701i)13-s + (−0.168 − 0.0941i)14-s + (−0.543 − 0.839i)16-s + 1.20i·17-s + 0.759i·19-s + (1.29 − 0.789i)20-s + (0.671 − 1.19i)22-s + (0.340 − 0.590i)23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.566 - 0.824i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.566 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$108$$    =    $$2^{2} \cdot 3^{3}$$ $$\varepsilon$$ = $0.566 - 0.824i$ motivic weight = $$3$$ character : $\chi_{108} (71, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 108,\ (\ :3/2),\ 0.566 - 0.824i)$$ $$L(2)$$ $$\approx$$ $$0.474658 + 0.249675i$$ $$L(\frac12)$$ $$\approx$$ $$0.474658 + 0.249675i$$ $$L(\frac{5}{2})$$ 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 + (1.44 + 2.43i)T$$
3 $$1$$
good5 $$1 + (14.6 + 8.45i)T + (62.5 + 108. i)T^{2}$$
7 $$1 + (-3.08 + 1.78i)T + (171.5 - 297. i)T^{2}$$
11 $$1 + (-25.0 - 43.4i)T + (-665.5 + 1.15e3i)T^{2}$$
13 $$1 + (18.9 - 32.9i)T + (-1.09e3 - 1.90e3i)T^{2}$$
17 $$1 - 84.3iT - 4.91e3T^{2}$$
19 $$1 - 62.9iT - 6.85e3T^{2}$$
23 $$1 + (-37.6 + 65.1i)T + (-6.08e3 - 1.05e4i)T^{2}$$
29 $$1 + (105. - 60.9i)T + (1.21e4 - 2.11e4i)T^{2}$$
31 $$1 + (-17.2 - 9.97i)T + (1.48e4 + 2.57e4i)T^{2}$$
37 $$1 - 17.7T + 5.06e4T^{2}$$
41 $$1 + (299. + 172. i)T + (3.44e4 + 5.96e4i)T^{2}$$
43 $$1 + (113. - 65.3i)T + (3.97e4 - 6.88e4i)T^{2}$$
47 $$1 + (153. + 265. i)T + (-5.19e4 + 8.99e4i)T^{2}$$
53 $$1 - 479. iT - 1.48e5T^{2}$$
59 $$1 + (245. - 425. i)T + (-1.02e5 - 1.77e5i)T^{2}$$
61 $$1 + (49.9 + 86.4i)T + (-1.13e5 + 1.96e5i)T^{2}$$
67 $$1 + (-536. - 309. i)T + (1.50e5 + 2.60e5i)T^{2}$$
71 $$1 - 254.T + 3.57e5T^{2}$$
73 $$1 - 100.T + 3.89e5T^{2}$$
79 $$1 + (856. - 494. i)T + (2.46e5 - 4.26e5i)T^{2}$$
83 $$1 + (-251. - 436. i)T + (-2.85e5 + 4.95e5i)T^{2}$$
89 $$1 + 1.01e3iT - 7.04e5T^{2}$$
97 $$1 + (503. + 872. i)T + (-4.56e5 + 7.90e5i)T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}