# Properties

 Degree 2 Conductor $2^{4} \cdot 3^{2} \cdot 5$ Sign $0.836 - 0.548i$ Motivic weight 5 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−46.7 + 30.6i)5-s − 179. i·7-s − 653.·11-s + 284. i·13-s − 383. i·17-s − 2.56e3·19-s − 948. i·23-s + (1.24e3 − 2.86e3i)25-s − 1.52e3·29-s − 3.10e3·31-s + (5.51e3 + 8.40e3i)35-s + 9.99e3i·37-s − 1.51e4·41-s − 1.75e3i·43-s − 1.47e4i·47-s + ⋯
 L(s)  = 1 + (−0.836 + 0.548i)5-s − 1.38i·7-s − 1.62·11-s + 0.467i·13-s − 0.321i·17-s − 1.62·19-s − 0.373i·23-s + (0.398 − 0.917i)25-s − 0.336·29-s − 0.580·31-s + (0.761 + 1.16i)35-s + 1.19i·37-s − 1.40·41-s − 0.144i·43-s − 0.974i·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.836 - 0.548i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.836 - 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$720$$    =    $$2^{4} \cdot 3^{2} \cdot 5$$ $$\varepsilon$$ = $0.836 - 0.548i$ motivic weight = $$5$$ character : $\chi_{720} (289, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 720,\ (\ :5/2),\ 0.836 - 0.548i)$$ $$L(3)$$ $$\approx$$ $$0.5615320340$$ $$L(\frac12)$$ $$\approx$$ $$0.5615320340$$ $$L(\frac{7}{2})$$ 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 + (46.7 - 30.6i)T$$
good7 $$1 + 179. iT - 1.68e4T^{2}$$
11 $$1 + 653.T + 1.61e5T^{2}$$
13 $$1 - 284. iT - 3.71e5T^{2}$$
17 $$1 + 383. iT - 1.41e6T^{2}$$
19 $$1 + 2.56e3T + 2.47e6T^{2}$$
23 $$1 + 948. iT - 6.43e6T^{2}$$
29 $$1 + 1.52e3T + 2.05e7T^{2}$$
31 $$1 + 3.10e3T + 2.86e7T^{2}$$
37 $$1 - 9.99e3iT - 6.93e7T^{2}$$
41 $$1 + 1.51e4T + 1.15e8T^{2}$$
43 $$1 + 1.75e3iT - 1.47e8T^{2}$$
47 $$1 + 1.47e4iT - 2.29e8T^{2}$$
53 $$1 - 8.70e3iT - 4.18e8T^{2}$$
59 $$1 - 1.26e4T + 7.14e8T^{2}$$
61 $$1 - 4.30e4T + 8.44e8T^{2}$$
67 $$1 + 2.61e4iT - 1.35e9T^{2}$$
71 $$1 + 4.62e4T + 1.80e9T^{2}$$
73 $$1 + 5.13e4iT - 2.07e9T^{2}$$
79 $$1 - 3.93e4T + 3.07e9T^{2}$$
83 $$1 + 6.95e4iT - 3.93e9T^{2}$$
89 $$1 + 1.30e4T + 5.58e9T^{2}$$
97 $$1 - 2.62e4iT - 8.58e9T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}