# Properties

 Degree 2 Conductor $2^{4} \cdot 5$ Sign $-0.0894 + 0.995i$ Motivic weight 5 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 11.1i·3-s + (−5 + 55.6i)5-s − 122. i·7-s + 119.·9-s + 100·11-s − 734. i·13-s + (619. + 55.6i)15-s − 979. i·17-s − 2.24e3·19-s − 1.36e3·21-s − 3.41e3i·23-s + (−3.07e3 − 556. i)25-s − 4.03e3i·27-s + 7.85e3·29-s + 2.14e3·31-s + ⋯
 L(s)  = 1 − 0.714i·3-s + (−0.0894 + 0.995i)5-s − 0.944i·7-s + 0.489·9-s + 0.249·11-s − 1.20i·13-s + (0.711 + 0.0638i)15-s − 0.822i·17-s − 1.42·19-s − 0.674·21-s − 1.34i·23-s + (−0.983 − 0.178i)25-s − 1.06i·27-s + 1.73·29-s + 0.400·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0894 + 0.995i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0894 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$80$$    =    $$2^{4} \cdot 5$$ $$\varepsilon$$ = $-0.0894 + 0.995i$ motivic weight = $$5$$ character : $\chi_{80} (49, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 80,\ (\ :5/2),\ -0.0894 + 0.995i)$$ $$L(3)$$ $$\approx$$ $$1.04981 - 1.14831i$$ $$L(\frac12)$$ $$\approx$$ $$1.04981 - 1.14831i$$ $$L(\frac{7}{2})$$ 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 + (5 - 55.6i)T$$
good3 $$1 + 11.1iT - 243T^{2}$$
7 $$1 + 122. iT - 1.68e4T^{2}$$
11 $$1 - 100T + 1.61e5T^{2}$$
13 $$1 + 734. iT - 3.71e5T^{2}$$
17 $$1 + 979. iT - 1.41e6T^{2}$$
19 $$1 + 2.24e3T + 2.47e6T^{2}$$
23 $$1 + 3.41e3iT - 6.43e6T^{2}$$
29 $$1 - 7.85e3T + 2.05e7T^{2}$$
31 $$1 - 2.14e3T + 2.86e7T^{2}$$
37 $$1 + 1.04e4iT - 6.93e7T^{2}$$
41 $$1 + 7.41e3T + 1.15e8T^{2}$$
43 $$1 - 1.77e4iT - 1.47e8T^{2}$$
47 $$1 + 9.43e3iT - 2.29e8T^{2}$$
53 $$1 - 2.42e4iT - 4.18e8T^{2}$$
59 $$1 - 2.59e4T + 7.14e8T^{2}$$
61 $$1 + 3.05e3T + 8.44e8T^{2}$$
67 $$1 - 5.87e4iT - 1.35e9T^{2}$$
71 $$1 + 3.76e4T + 1.80e9T^{2}$$
73 $$1 + 2.40e4iT - 2.07e9T^{2}$$
79 $$1 - 7.97e4T + 3.07e9T^{2}$$
83 $$1 - 1.62e4iT - 3.93e9T^{2}$$
89 $$1 - 826T + 5.58e9T^{2}$$
97 $$1 - 3.75e4iT - 8.58e9T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}