# Properties

 Degree $2$ Conductor $160$ Sign $0.960 - 0.277i$ Motivic weight $5$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−13.4 − 13.4i)3-s + (15.0 − 53.8i)5-s + (−76.4 + 76.4i)7-s + 117. i·9-s + 622. i·11-s + (−293. + 293. i)13-s + (−925. + 521. i)15-s + (−1.15e3 − 1.15e3i)17-s + 2.00e3·19-s + 2.05e3·21-s + (1.39e3 + 1.39e3i)23-s + (−2.67e3 − 1.61e3i)25-s + (−1.68e3 + 1.68e3i)27-s − 305. i·29-s − 2.10e3i·31-s + ⋯
 L(s)  = 1 + (−0.861 − 0.861i)3-s + (0.268 − 0.963i)5-s + (−0.589 + 0.589i)7-s + 0.485i·9-s + 1.55i·11-s + (−0.481 + 0.481i)13-s + (−1.06 + 0.598i)15-s + (−0.967 − 0.967i)17-s + 1.27·19-s + 1.01·21-s + (0.548 + 0.548i)23-s + (−0.855 − 0.518i)25-s + (−0.443 + 0.443i)27-s − 0.0675i·29-s − 0.393i·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 - 0.277i)\, \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 - 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$160$$    =    $$2^{5} \cdot 5$$ Sign: $0.960 - 0.277i$ Motivic weight: $$5$$ Character: $\chi_{160} (127, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 160,\ (\ :5/2),\ 0.960 - 0.277i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$0.9786008140$$ $$L(\frac12)$$ $$\approx$$ $$0.9786008140$$ $$L(\frac{7}{2})$$ 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 + (-15.0 + 53.8i)T$$
good3 $$1 + (13.4 + 13.4i)T + 243iT^{2}$$
7 $$1 + (76.4 - 76.4i)T - 1.68e4iT^{2}$$
11 $$1 - 622. iT - 1.61e5T^{2}$$
13 $$1 + (293. - 293. i)T - 3.71e5iT^{2}$$
17 $$1 + (1.15e3 + 1.15e3i)T + 1.41e6iT^{2}$$
19 $$1 - 2.00e3T + 2.47e6T^{2}$$
23 $$1 + (-1.39e3 - 1.39e3i)T + 6.43e6iT^{2}$$
29 $$1 + 305. iT - 2.05e7T^{2}$$
31 $$1 + 2.10e3iT - 2.86e7T^{2}$$
37 $$1 + (-9.90e3 - 9.90e3i)T + 6.93e7iT^{2}$$
41 $$1 - 1.69e4T + 1.15e8T^{2}$$
43 $$1 + (2.03e3 + 2.03e3i)T + 1.47e8iT^{2}$$
47 $$1 + (682. - 682. i)T - 2.29e8iT^{2}$$
53 $$1 + (-2.10e4 + 2.10e4i)T - 4.18e8iT^{2}$$
59 $$1 - 1.16e4T + 7.14e8T^{2}$$
61 $$1 + 1.30e3T + 8.44e8T^{2}$$
67 $$1 + (3.98e4 - 3.98e4i)T - 1.35e9iT^{2}$$
71 $$1 + 2.54e4iT - 1.80e9T^{2}$$
73 $$1 + (-1.03e3 + 1.03e3i)T - 2.07e9iT^{2}$$
79 $$1 + 1.18e4T + 3.07e9T^{2}$$
83 $$1 + (-4.51e4 - 4.51e4i)T + 3.93e9iT^{2}$$
89 $$1 - 1.43e5iT - 5.58e9T^{2}$$
97 $$1 + (2.43e4 + 2.43e4i)T + 8.58e9iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$