# Properties

 Degree 2 Conductor $2^{4} \cdot 3^{2} \cdot 5$ Sign $-0.235 + 0.971i$ Motivic weight 5 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (13.1 − 54.3i)5-s + 146. i·7-s + 191.·11-s + 83.9i·13-s − 2.00e3i·17-s + 677.·19-s + 1.29e3i·23-s + (−2.77e3 − 1.42e3i)25-s − 3.26e3·29-s − 6.15e3·31-s + (7.97e3 + 1.93e3i)35-s + 1.13e4i·37-s + 1.05e4·41-s − 1.29e4i·43-s − 9.52e3i·47-s + ⋯
 L(s)  = 1 + (0.235 − 0.971i)5-s + 1.13i·7-s + 0.476·11-s + 0.137i·13-s − 1.67i·17-s + 0.430·19-s + 0.510i·23-s + (−0.889 − 0.457i)25-s − 0.721·29-s − 1.15·31-s + (1.10 + 0.266i)35-s + 1.36i·37-s + 0.984·41-s − 1.06i·43-s − 0.628i·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.235 + 0.971i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.235 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$720$$    =    $$2^{4} \cdot 3^{2} \cdot 5$$ $$\varepsilon$$ = $-0.235 + 0.971i$ motivic weight = $$5$$ character : $\chi_{720} (289, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 720,\ (\ :5/2),\ -0.235 + 0.971i)$$ $$L(3)$$ $$\approx$$ $$1.646412068$$ $$L(\frac12)$$ $$\approx$$ $$1.646412068$$ $$L(\frac{7}{2})$$ 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 + (-13.1 + 54.3i)T$$
good7 $$1 - 146. iT - 1.68e4T^{2}$$
11 $$1 - 191.T + 1.61e5T^{2}$$
13 $$1 - 83.9iT - 3.71e5T^{2}$$
17 $$1 + 2.00e3iT - 1.41e6T^{2}$$
19 $$1 - 677.T + 2.47e6T^{2}$$
23 $$1 - 1.29e3iT - 6.43e6T^{2}$$
29 $$1 + 3.26e3T + 2.05e7T^{2}$$
31 $$1 + 6.15e3T + 2.86e7T^{2}$$
37 $$1 - 1.13e4iT - 6.93e7T^{2}$$
41 $$1 - 1.05e4T + 1.15e8T^{2}$$
43 $$1 + 1.29e4iT - 1.47e8T^{2}$$
47 $$1 + 9.52e3iT - 2.29e8T^{2}$$
53 $$1 + 1.47e4iT - 4.18e8T^{2}$$
59 $$1 - 3.82e4T + 7.14e8T^{2}$$
61 $$1 + 3.58e3T + 8.44e8T^{2}$$
67 $$1 - 2.17e4iT - 1.35e9T^{2}$$
71 $$1 - 5.13e4T + 1.80e9T^{2}$$
73 $$1 + 1.33e4iT - 2.07e9T^{2}$$
79 $$1 - 1.59e4T + 3.07e9T^{2}$$
83 $$1 + 5.33e4iT - 3.93e9T^{2}$$
89 $$1 + 5.13e4T + 5.58e9T^{2}$$
97 $$1 + 8.08e4iT - 8.58e9T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}