# Properties

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

# Related objects

## Dirichlet series

 L(s)  = 1 + 28.9i·3-s + (−13.1 − 54.3i)5-s − 146. i·7-s − 594.·9-s − 191.·11-s − 83.9i·13-s + (1.57e3 − 380. i)15-s − 2.00e3i·17-s + 677.·19-s + 4.24e3·21-s + 1.29e3i·23-s + (−2.77e3 + 1.42e3i)25-s − 1.01e4i·27-s + 3.26e3·29-s − 6.15e3·31-s + ⋯
 L(s)  = 1 + 1.85i·3-s + (−0.235 − 0.971i)5-s − 1.13i·7-s − 2.44·9-s − 0.476·11-s − 0.137i·13-s + (1.80 − 0.436i)15-s − 1.67i·17-s + 0.430·19-s + 2.10·21-s + 0.510i·23-s + (−0.889 + 0.457i)25-s − 2.68i·27-s + 0.721·29-s − 1.15·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.235 + 0.971i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.235 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$80$$    =    $$2^{4} \cdot 5$$ $$\varepsilon$$ = $0.235 + 0.971i$ motivic weight = $$5$$ character : $\chi_{80} (49, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 80,\ (\ :5/2),\ 0.235 + 0.971i)$$ $$L(3)$$ $$\approx$$ $$0.615601 - 0.484302i$$ $$L(\frac12)$$ $$\approx$$ $$0.615601 - 0.484302i$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;5\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
5 $$1 + (13.1 + 54.3i)T$$
good3 $$1 - 28.9iT - 243T^{2}$$
7 $$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}