# Properties

 Degree 2 Conductor $2^{2} \cdot 3^{3}$ Sign $0.730 + 0.682i$ Motivic weight 4 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.59 − 3.66i)2-s + (−10.9 + 11.6i)4-s − 29.4·5-s + 32.9i·7-s + (60.2 + 21.4i)8-s + (46.9 + 108. i)10-s − 136. i·11-s + 164.·13-s + (120. − 52.4i)14-s + (−17.2 − 255. i)16-s + 380.·17-s + 439. i·19-s + (321. − 344. i)20-s + (−501. + 217. i)22-s − 171. i·23-s + ⋯
 L(s)  = 1 + (−0.398 − 0.917i)2-s + (−0.682 + 0.730i)4-s − 1.17·5-s + 0.671i·7-s + (0.942 + 0.335i)8-s + (0.469 + 1.08i)10-s − 1.12i·11-s + 0.973·13-s + (0.616 − 0.267i)14-s + (−0.0672 − 0.997i)16-s + 1.31·17-s + 1.21i·19-s + (0.804 − 0.860i)20-s + (−1.03 + 0.449i)22-s − 0.324i·23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.730 + 0.682i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.730 + 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$108$$    =    $$2^{2} \cdot 3^{3}$$ $$\varepsilon$$ = $0.730 + 0.682i$ motivic weight = $$4$$ character : $\chi_{108} (55, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 108,\ (\ :2),\ 0.730 + 0.682i)$$ $$L(\frac{5}{2})$$ $$\approx$$ $$0.987674 - 0.389780i$$ $$L(\frac12)$$ $$\approx$$ $$0.987674 - 0.389780i$$ $$L(3)$$ 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 + (1.59 + 3.66i)T$$
3 $$1$$
good5 $$1 + 29.4T + 625T^{2}$$
7 $$1 - 32.9iT - 2.40e3T^{2}$$
11 $$1 + 136. iT - 1.46e4T^{2}$$
13 $$1 - 164.T + 2.85e4T^{2}$$
17 $$1 - 380.T + 8.35e4T^{2}$$
19 $$1 - 439. iT - 1.30e5T^{2}$$
23 $$1 + 171. iT - 2.79e5T^{2}$$
29 $$1 - 1.04e3T + 7.07e5T^{2}$$
31 $$1 + 1.18e3iT - 9.23e5T^{2}$$
37 $$1 - 2.70e3T + 1.87e6T^{2}$$
41 $$1 + 1.55e3T + 2.82e6T^{2}$$
43 $$1 - 2.21e3iT - 3.41e6T^{2}$$
47 $$1 - 1.29e3iT - 4.87e6T^{2}$$
53 $$1 + 1.01e3T + 7.89e6T^{2}$$
59 $$1 - 2.43e3iT - 1.21e7T^{2}$$
61 $$1 - 3.83e3T + 1.38e7T^{2}$$
67 $$1 - 2.35e3iT - 2.01e7T^{2}$$
71 $$1 + 884. iT - 2.54e7T^{2}$$
73 $$1 - 6.92e3T + 2.83e7T^{2}$$
79 $$1 + 1.03e4iT - 3.89e7T^{2}$$
83 $$1 - 1.23e4iT - 4.74e7T^{2}$$
89 $$1 - 5.75e3T + 6.27e7T^{2}$$
97 $$1 + 8.15e3T + 8.85e7T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}