# Properties

 Degree $2$ Conductor $160$ Sign $0.950 - 0.310i$ Motivic weight $5$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (9.28 − 9.28i)3-s + (42.6 + 36.0i)5-s + (−105. − 105. i)7-s + 70.7i·9-s + 344. i·11-s + (707. + 707. i)13-s + (731. − 61.3i)15-s + (1.26e3 − 1.26e3i)17-s + 438.·19-s − 1.96e3·21-s + (−1.72e3 + 1.72e3i)23-s + (520. + 3.08e3i)25-s + (2.91e3 + 2.91e3i)27-s + 2.51e3i·29-s − 7.14e3i·31-s + ⋯
 L(s)  = 1 + (0.595 − 0.595i)3-s + (0.763 + 0.645i)5-s + (−0.816 − 0.816i)7-s + 0.291i·9-s + 0.859i·11-s + (1.16 + 1.16i)13-s + (0.839 − 0.0703i)15-s + (1.05 − 1.05i)17-s + 0.278·19-s − 0.971·21-s + (−0.679 + 0.679i)23-s + (0.166 + 0.986i)25-s + (0.768 + 0.768i)27-s + 0.554i·29-s − 1.33i·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 - 0.310i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.950 - 0.310i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$160$$    =    $$2^{5} \cdot 5$$ Sign: $0.950 - 0.310i$ Motivic weight: $$5$$ Character: $\chi_{160} (63, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 160,\ (\ :5/2),\ 0.950 - 0.310i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$2.658406422$$ $$L(\frac12)$$ $$\approx$$ $$2.658406422$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$1 + (-42.6 - 36.0i)T$$
good3 $$1 + (-9.28 + 9.28i)T - 243iT^{2}$$
7 $$1 + (105. + 105. i)T + 1.68e4iT^{2}$$
11 $$1 - 344. iT - 1.61e5T^{2}$$
13 $$1 + (-707. - 707. i)T + 3.71e5iT^{2}$$
17 $$1 + (-1.26e3 + 1.26e3i)T - 1.41e6iT^{2}$$
19 $$1 - 438.T + 2.47e6T^{2}$$
23 $$1 + (1.72e3 - 1.72e3i)T - 6.43e6iT^{2}$$
29 $$1 - 2.51e3iT - 2.05e7T^{2}$$
31 $$1 + 7.14e3iT - 2.86e7T^{2}$$
37 $$1 + (3.36e3 - 3.36e3i)T - 6.93e7iT^{2}$$
41 $$1 - 6.96e3T + 1.15e8T^{2}$$
43 $$1 + (-1.63e4 + 1.63e4i)T - 1.47e8iT^{2}$$
47 $$1 + (-1.30e4 - 1.30e4i)T + 2.29e8iT^{2}$$
53 $$1 + (-2.00e4 - 2.00e4i)T + 4.18e8iT^{2}$$
59 $$1 - 8.41e3T + 7.14e8T^{2}$$
61 $$1 + 2.29e3T + 8.44e8T^{2}$$
67 $$1 + (-395. - 395. i)T + 1.35e9iT^{2}$$
71 $$1 - 4.08e4iT - 1.80e9T^{2}$$
73 $$1 + (4.56e4 + 4.56e4i)T + 2.07e9iT^{2}$$
79 $$1 + 5.84e4T + 3.07e9T^{2}$$
83 $$1 + (-2.69e4 + 2.69e4i)T - 3.93e9iT^{2}$$
89 $$1 - 6.19e4iT - 5.58e9T^{2}$$
97 $$1 + (1.10e4 - 1.10e4i)T - 8.58e9iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$