# Properties

 Degree 2 Conductor $2^{5} \cdot 5$ 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

 $$d$$ = $$2$$ $$N$$ = $$160$$    =    $$2^{5} \cdot 5$$ $$\varepsilon$$ = $0.960 + 0.277i$ motivic weight = $$5$$ character : $\chi_{160} (63, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 160,\ (\ :5/2),\ 0.960 + 0.277i)$$ $$L(3)$$ $$\approx$$ $$0.9786008140$$ $$L(\frac12)$$ $$\approx$$ $$0.9786008140$$ $$L(\frac{7}{2})$$ 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 + (-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}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}