# Properties

 Degree 2 Conductor $2^{4} \cdot 3^{2} \cdot 5$ Sign $-0.983 + 0.178i$ Motivic weight 5 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−55 + 10i)5-s − 158i·7-s − 148·11-s − 684i·13-s − 2.04e3i·17-s + 2.22e3·19-s + 1.24e3i·23-s + (2.92e3 − 1.10e3i)25-s − 270·29-s + 2.04e3·31-s + (1.58e3 + 8.69e3i)35-s − 4.37e3i·37-s + 2.39e3·41-s + 2.29e3i·43-s − 1.06e4i·47-s + ⋯
 L(s)  = 1 + (−0.983 + 0.178i)5-s − 1.21i·7-s − 0.368·11-s − 1.12i·13-s − 1.71i·17-s + 1.41·19-s + 0.491i·23-s + (0.936 − 0.352i)25-s − 0.0596·29-s + 0.382·31-s + (0.218 + 1.19i)35-s − 0.525i·37-s + 0.222·41-s + 0.189i·43-s − 0.705i·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.178i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.983 + 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$720$$    =    $$2^{4} \cdot 3^{2} \cdot 5$$ $$\varepsilon$$ = $-0.983 + 0.178i$ motivic weight = $$5$$ character : $\chi_{720} (289, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 720,\ (\ :5/2),\ -0.983 + 0.178i)$$ $$L(3)$$ $$\approx$$ $$1.123461028$$ $$L(\frac12)$$ $$\approx$$ $$1.123461028$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;5\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
5 $$1 + (55 - 10i)T$$
good7 $$1 + 158iT - 1.68e4T^{2}$$
11 $$1 + 148T + 1.61e5T^{2}$$
13 $$1 + 684iT - 3.71e5T^{2}$$
17 $$1 + 2.04e3iT - 1.41e6T^{2}$$
19 $$1 - 2.22e3T + 2.47e6T^{2}$$
23 $$1 - 1.24e3iT - 6.43e6T^{2}$$
29 $$1 + 270T + 2.05e7T^{2}$$
31 $$1 - 2.04e3T + 2.86e7T^{2}$$
37 $$1 + 4.37e3iT - 6.93e7T^{2}$$
41 $$1 - 2.39e3T + 1.15e8T^{2}$$
43 $$1 - 2.29e3iT - 1.47e8T^{2}$$
47 $$1 + 1.06e4iT - 2.29e8T^{2}$$
53 $$1 - 2.96e3iT - 4.18e8T^{2}$$
59 $$1 - 3.97e4T + 7.14e8T^{2}$$
61 $$1 + 4.22e4T + 8.44e8T^{2}$$
67 $$1 + 3.20e4iT - 1.35e9T^{2}$$
71 $$1 + 4.24e3T + 1.80e9T^{2}$$
73 $$1 + 3.01e4iT - 2.07e9T^{2}$$
79 $$1 - 3.52e4T + 3.07e9T^{2}$$
83 $$1 - 2.78e4iT - 3.93e9T^{2}$$
89 $$1 + 8.52e4T + 5.58e9T^{2}$$
97 $$1 + 9.72e4iT - 8.58e9T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}