# Properties

 Degree 2 Conductor $2^{2} \cdot 3^{3}$ Sign $0.574 - 0.818i$ Motivic weight 4 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (3.81 + 1.20i)2-s + (13.0 + 9.19i)4-s + 13.3·5-s + 25.4i·7-s + (38.8 + 50.8i)8-s + (51.0 + 16.1i)10-s + 34.7i·11-s + 116.·13-s + (−30.6 + 96.9i)14-s + (86.9 + 240. i)16-s + 109.·17-s − 264. i·19-s + (175. + 123. i)20-s + (−41.8 + 132. i)22-s + 890. i·23-s + ⋯
 L(s)  = 1 + (0.953 + 0.301i)2-s + (0.818 + 0.574i)4-s + 0.535·5-s + 0.518i·7-s + (0.607 + 0.794i)8-s + (0.510 + 0.161i)10-s + 0.287i·11-s + 0.691·13-s + (−0.156 + 0.494i)14-s + (0.339 + 0.940i)16-s + 0.379·17-s − 0.733i·19-s + (0.438 + 0.307i)20-s + (−0.0865 + 0.273i)22-s + 1.68i·23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.574 - 0.818i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.574 - 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$108$$    =    $$2^{2} \cdot 3^{3}$$ $$\varepsilon$$ = $0.574 - 0.818i$ motivic weight = $$4$$ character : $\chi_{108} (55, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 108,\ (\ :2),\ 0.574 - 0.818i)$$ $$L(\frac{5}{2})$$ $$\approx$$ $$3.04284 + 1.58151i$$ $$L(\frac12)$$ $$\approx$$ $$3.04284 + 1.58151i$$ $$L(3)$$ 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 + (-3.81 - 1.20i)T$$
3 $$1$$
good5 $$1 - 13.3T + 625T^{2}$$
7 $$1 - 25.4iT - 2.40e3T^{2}$$
11 $$1 - 34.7iT - 1.46e4T^{2}$$
13 $$1 - 116.T + 2.85e4T^{2}$$
17 $$1 - 109.T + 8.35e4T^{2}$$
19 $$1 + 264. iT - 1.30e5T^{2}$$
23 $$1 - 890. iT - 2.79e5T^{2}$$
29 $$1 - 1.11e3T + 7.07e5T^{2}$$
31 $$1 + 1.66e3iT - 9.23e5T^{2}$$
37 $$1 + 1.98e3T + 1.87e6T^{2}$$
41 $$1 + 1.22e3T + 2.82e6T^{2}$$
43 $$1 + 3.14e3iT - 3.41e6T^{2}$$
47 $$1 + 2.36e3iT - 4.87e6T^{2}$$
53 $$1 + 2.18e3T + 7.89e6T^{2}$$
59 $$1 + 1.80e3iT - 1.21e7T^{2}$$
61 $$1 - 1.51e3T + 1.38e7T^{2}$$
67 $$1 + 5.25e3iT - 2.01e7T^{2}$$
71 $$1 + 1.12e3iT - 2.54e7T^{2}$$
73 $$1 + 6.35e3T + 2.83e7T^{2}$$
79 $$1 - 445. iT - 3.89e7T^{2}$$
83 $$1 - 1.11e3iT - 4.74e7T^{2}$$
89 $$1 + 1.03e4T + 6.27e7T^{2}$$
97 $$1 - 474.T + 8.85e7T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}