# Properties

 Degree 2 Conductor $2^{5} \cdot 5$ Sign $-0.327 + 0.944i$ Motivic weight 5 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−5.65 + 5.65i)3-s + (−54.5 + 12.2i)5-s + (23.8 + 23.8i)7-s + 178. i·9-s + 218. i·11-s + (152. + 152. i)13-s + (239. − 377. i)15-s + (318. − 318. i)17-s − 2.45e3·19-s − 270.·21-s + (−512. + 512. i)23-s + (2.82e3 − 1.33e3i)25-s + (−2.38e3 − 2.38e3i)27-s − 5.94e3i·29-s − 3.06e3i·31-s + ⋯
 L(s)  = 1 + (−0.362 + 0.362i)3-s + (−0.975 + 0.218i)5-s + (0.184 + 0.184i)7-s + 0.736i·9-s + 0.543i·11-s + (0.250 + 0.250i)13-s + (0.274 − 0.433i)15-s + (0.267 − 0.267i)17-s − 1.56·19-s − 0.133·21-s + (−0.201 + 0.201i)23-s + (0.904 − 0.426i)25-s + (−0.630 − 0.630i)27-s − 1.31i·29-s − 0.572i·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.327 + 0.944i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.327 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$160$$    =    $$2^{5} \cdot 5$$ $$\varepsilon$$ = $-0.327 + 0.944i$ motivic weight = $$5$$ character : $\chi_{160} (63, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 160,\ (\ :5/2),\ -0.327 + 0.944i)$$ $$L(3)$$ $$\approx$$ $$0.2031982984$$ $$L(\frac12)$$ $$\approx$$ $$0.2031982984$$ $$L(\frac{7}{2})$$ 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 + (54.5 - 12.2i)T$$
good3 $$1 + (5.65 - 5.65i)T - 243iT^{2}$$
7 $$1 + (-23.8 - 23.8i)T + 1.68e4iT^{2}$$
11 $$1 - 218. iT - 1.61e5T^{2}$$
13 $$1 + (-152. - 152. i)T + 3.71e5iT^{2}$$
17 $$1 + (-318. + 318. i)T - 1.41e6iT^{2}$$
19 $$1 + 2.45e3T + 2.47e6T^{2}$$
23 $$1 + (512. - 512. i)T - 6.43e6iT^{2}$$
29 $$1 + 5.94e3iT - 2.05e7T^{2}$$
31 $$1 + 3.06e3iT - 2.86e7T^{2}$$
37 $$1 + (-1.31e3 + 1.31e3i)T - 6.93e7iT^{2}$$
41 $$1 - 6.09e3T + 1.15e8T^{2}$$
43 $$1 + (1.90e3 - 1.90e3i)T - 1.47e8iT^{2}$$
47 $$1 + (8.04e3 + 8.04e3i)T + 2.29e8iT^{2}$$
53 $$1 + (8.09e3 + 8.09e3i)T + 4.18e8iT^{2}$$
59 $$1 + 4.16e4T + 7.14e8T^{2}$$
61 $$1 - 4.30e4T + 8.44e8T^{2}$$
67 $$1 + (4.16e4 + 4.16e4i)T + 1.35e9iT^{2}$$
71 $$1 + 2.37e4iT - 1.80e9T^{2}$$
73 $$1 + (-9.33e3 - 9.33e3i)T + 2.07e9iT^{2}$$
79 $$1 - 8.60e4T + 3.07e9T^{2}$$
83 $$1 + (7.58e4 - 7.58e4i)T - 3.93e9iT^{2}$$
89 $$1 + 1.86e4iT - 5.58e9T^{2}$$
97 $$1 + (9.77e4 - 9.77e4i)T - 8.58e9iT^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}