# Properties

 Degree $2$ Conductor $720$ Sign $-0.948 + 0.317i$ Motivic weight $5$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (53.0 − 17.7i)5-s − 188. i·7-s − 501.·11-s + 1.06e3i·13-s − 29.5i·17-s + 1.57e3·19-s − 1.29e3i·23-s + (2.49e3 − 1.88e3i)25-s − 3.58e3·29-s + 3.52e3·31-s + (−3.35e3 − 1.00e4i)35-s − 8.41e3i·37-s − 7.01e3·41-s − 2.26e4i·43-s + 3.50e3i·47-s + ⋯
 L(s)  = 1 + (0.948 − 0.317i)5-s − 1.45i·7-s − 1.25·11-s + 1.74i·13-s − 0.0248i·17-s + 1.00·19-s − 0.510i·23-s + (0.798 − 0.602i)25-s − 0.791·29-s + 0.659·31-s + (−0.463 − 1.38i)35-s − 1.01i·37-s − 0.651·41-s − 1.87i·43-s + 0.231i·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 + 0.317i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.948 + 0.317i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$720$$    =    $$2^{4} \cdot 3^{2} \cdot 5$$ Sign: $-0.948 + 0.317i$ Motivic weight: $$5$$ Character: $\chi_{720} (289, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 720,\ (\ :5/2),\ -0.948 + 0.317i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$1.138196341$$ $$L(\frac12)$$ $$\approx$$ $$1.138196341$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
5 $$1 + (-53.0 + 17.7i)T$$
good7 $$1 + 188. iT - 1.68e4T^{2}$$
11 $$1 + 501.T + 1.61e5T^{2}$$
13 $$1 - 1.06e3iT - 3.71e5T^{2}$$
17 $$1 + 29.5iT - 1.41e6T^{2}$$
19 $$1 - 1.57e3T + 2.47e6T^{2}$$
23 $$1 + 1.29e3iT - 6.43e6T^{2}$$
29 $$1 + 3.58e3T + 2.05e7T^{2}$$
31 $$1 - 3.52e3T + 2.86e7T^{2}$$
37 $$1 + 8.41e3iT - 6.93e7T^{2}$$
41 $$1 + 7.01e3T + 1.15e8T^{2}$$
43 $$1 + 2.26e4iT - 1.47e8T^{2}$$
47 $$1 - 3.50e3iT - 2.29e8T^{2}$$
53 $$1 + 2.73e4iT - 4.18e8T^{2}$$
59 $$1 + 7.92e3T + 7.14e8T^{2}$$
61 $$1 + 7.02e3T + 8.44e8T^{2}$$
67 $$1 - 1.76e4iT - 1.35e9T^{2}$$
71 $$1 - 1.34e4T + 1.80e9T^{2}$$
73 $$1 - 3.99e4iT - 2.07e9T^{2}$$
79 $$1 + 9.33e4T + 3.07e9T^{2}$$
83 $$1 + 5.84e4iT - 3.93e9T^{2}$$
89 $$1 + 1.39e4T + 5.58e9T^{2}$$
97 $$1 - 1.10e5iT - 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$