# Properties

 Degree $2$ Conductor $160$ Sign $-0.692 - 0.721i$ Motivic weight $5$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−16.5 + 16.5i)3-s + (13.9 − 54.1i)5-s + (−2.15 − 2.15i)7-s − 304. i·9-s + 255. i·11-s + (111. + 111. i)13-s + (665. + 1.12e3i)15-s + (998. − 998. i)17-s + 1.94e3·19-s + 71.4·21-s + (−1.61e3 + 1.61e3i)23-s + (−2.73e3 − 1.50e3i)25-s + (1.02e3 + 1.02e3i)27-s + 5.84e3i·29-s + 1.60e3i·31-s + ⋯
 L(s)  = 1 + (−1.06 + 1.06i)3-s + (0.249 − 0.968i)5-s + (−0.0166 − 0.0166i)7-s − 1.25i·9-s + 0.637i·11-s + (0.183 + 0.183i)13-s + (0.763 + 1.29i)15-s + (0.838 − 0.838i)17-s + 1.23·19-s + 0.0353·21-s + (−0.636 + 0.636i)23-s + (−0.875 − 0.482i)25-s + (0.270 + 0.270i)27-s + 1.29i·29-s + 0.300i·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.692 - 0.721i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.692 - 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$160$$    =    $$2^{5} \cdot 5$$ Sign: $-0.692 - 0.721i$ Motivic weight: $$5$$ Character: $\chi_{160} (63, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 160,\ (\ :5/2),\ -0.692 - 0.721i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$0.8207110257$$ $$L(\frac12)$$ $$\approx$$ $$0.8207110257$$ $$L(\frac{7}{2})$$ 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 + (-13.9 + 54.1i)T$$
good3 $$1 + (16.5 - 16.5i)T - 243iT^{2}$$
7 $$1 + (2.15 + 2.15i)T + 1.68e4iT^{2}$$
11 $$1 - 255. iT - 1.61e5T^{2}$$
13 $$1 + (-111. - 111. i)T + 3.71e5iT^{2}$$
17 $$1 + (-998. + 998. i)T - 1.41e6iT^{2}$$
19 $$1 - 1.94e3T + 2.47e6T^{2}$$
23 $$1 + (1.61e3 - 1.61e3i)T - 6.43e6iT^{2}$$
29 $$1 - 5.84e3iT - 2.05e7T^{2}$$
31 $$1 - 1.60e3iT - 2.86e7T^{2}$$
37 $$1 + (1.13e4 - 1.13e4i)T - 6.93e7iT^{2}$$
41 $$1 + 8.43e3T + 1.15e8T^{2}$$
43 $$1 + (1.36e4 - 1.36e4i)T - 1.47e8iT^{2}$$
47 $$1 + (1.27e4 + 1.27e4i)T + 2.29e8iT^{2}$$
53 $$1 + (-2.03e3 - 2.03e3i)T + 4.18e8iT^{2}$$
59 $$1 - 2.75e4T + 7.14e8T^{2}$$
61 $$1 - 9.59e3T + 8.44e8T^{2}$$
67 $$1 + (-3.74e4 - 3.74e4i)T + 1.35e9iT^{2}$$
71 $$1 - 6.00e4iT - 1.80e9T^{2}$$
73 $$1 + (-2.50e4 - 2.50e4i)T + 2.07e9iT^{2}$$
79 $$1 + 4.92e4T + 3.07e9T^{2}$$
83 $$1 + (4.46e4 - 4.46e4i)T - 3.93e9iT^{2}$$
89 $$1 + 2.04e4iT - 5.58e9T^{2}$$
97 $$1 + (1.00e5 - 1.00e5i)T - 8.58e9iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$