# Properties

 Degree 2 Conductor $2^{5} \cdot 5$ Sign $-0.960 - 0.279i$ Motivic weight 5 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (10.9 − 10.9i)3-s + (−14.9 − 53.8i)5-s + (−75.2 − 75.2i)7-s + 2.76i·9-s − 207. i·11-s + (−233. − 233. i)13-s + (−753. − 426. i)15-s + (−721. + 721. i)17-s + 114.·19-s − 1.64e3·21-s + (−957. + 957. i)23-s + (−2.67e3 + 1.60e3i)25-s + (2.69e3 + 2.69e3i)27-s − 4.18e3i·29-s + 2.74e3i·31-s + ⋯
 L(s)  = 1 + (0.703 − 0.703i)3-s + (−0.266 − 0.963i)5-s + (−0.580 − 0.580i)7-s + 0.0113i·9-s − 0.517i·11-s + (−0.383 − 0.383i)13-s + (−0.865 − 0.489i)15-s + (−0.605 + 0.605i)17-s + 0.0724·19-s − 0.816·21-s + (−0.377 + 0.377i)23-s + (−0.857 + 0.514i)25-s + (0.711 + 0.711i)27-s − 0.924i·29-s + 0.512i·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 - 0.279i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.960 - 0.279i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$160$$    =    $$2^{5} \cdot 5$$ $$\varepsilon$$ = $-0.960 - 0.279i$ motivic weight = $$5$$ character : $\chi_{160} (63, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 160,\ (\ :5/2),\ -0.960 - 0.279i)$$ $$L(3)$$ $$\approx$$ $$0.9695445655$$ $$L(\frac12)$$ $$\approx$$ $$0.9695445655$$ $$L(\frac{7}{2})$$ 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 + (14.9 + 53.8i)T$$
good3 $$1 + (-10.9 + 10.9i)T - 243iT^{2}$$
7 $$1 + (75.2 + 75.2i)T + 1.68e4iT^{2}$$
11 $$1 + 207. iT - 1.61e5T^{2}$$
13 $$1 + (233. + 233. i)T + 3.71e5iT^{2}$$
17 $$1 + (721. - 721. i)T - 1.41e6iT^{2}$$
19 $$1 - 114.T + 2.47e6T^{2}$$
23 $$1 + (957. - 957. i)T - 6.43e6iT^{2}$$
29 $$1 + 4.18e3iT - 2.05e7T^{2}$$
31 $$1 - 2.74e3iT - 2.86e7T^{2}$$
37 $$1 + (-6.90e3 + 6.90e3i)T - 6.93e7iT^{2}$$
41 $$1 + 8.45e3T + 1.15e8T^{2}$$
43 $$1 + (1.33e4 - 1.33e4i)T - 1.47e8iT^{2}$$
47 $$1 + (-6.67e3 - 6.67e3i)T + 2.29e8iT^{2}$$
53 $$1 + (1.30e4 + 1.30e4i)T + 4.18e8iT^{2}$$
59 $$1 + 1.13e4T + 7.14e8T^{2}$$
61 $$1 + 5.60e4T + 8.44e8T^{2}$$
67 $$1 + (-3.58e4 - 3.58e4i)T + 1.35e9iT^{2}$$
71 $$1 + 7.42e4iT - 1.80e9T^{2}$$
73 $$1 + (5.88e4 + 5.88e4i)T + 2.07e9iT^{2}$$
79 $$1 + 6.53e4T + 3.07e9T^{2}$$
83 $$1 + (-3.55e4 + 3.55e4i)T - 3.93e9iT^{2}$$
89 $$1 - 1.11e4iT - 5.58e9T^{2}$$
97 $$1 + (-8.66e4 + 8.66e4i)T - 8.58e9iT^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}