# Properties

 Degree 2 Conductor $2^{6} \cdot 5$ Sign $-0.707 - 0.707i$ Motivic weight 5 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 15.6i·3-s − 25i·5-s − 164.·7-s − 2.80·9-s − 744. i·11-s − 873. i·13-s − 391.·15-s + 984.·17-s + 1.14e3i·19-s + 2.58e3i·21-s + 578.·23-s − 625·25-s − 3.76e3i·27-s − 5.86e3i·29-s − 9.82e3·31-s + ⋯
 L(s)  = 1 − 1.00i·3-s − 0.447i·5-s − 1.27·7-s − 0.0115·9-s − 1.85i·11-s − 1.43i·13-s − 0.449·15-s + 0.826·17-s + 0.730i·19-s + 1.27i·21-s + 0.227·23-s − 0.200·25-s − 0.994i·27-s − 1.29i·29-s − 1.83·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$320$$    =    $$2^{6} \cdot 5$$ $$\varepsilon$$ = $-0.707 - 0.707i$ motivic weight = $$5$$ character : $\chi_{320} (161, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 320,\ (\ :5/2),\ -0.707 - 0.707i)$$ $$L(3)$$ $$\approx$$ $$1.062844095$$ $$L(\frac12)$$ $$\approx$$ $$1.062844095$$ $$L(\frac{7}{2})$$ 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 + 25iT$$
good3 $$1 + 15.6iT - 243T^{2}$$
7 $$1 + 164.T + 1.68e4T^{2}$$
11 $$1 + 744. iT - 1.61e5T^{2}$$
13 $$1 + 873. iT - 3.71e5T^{2}$$
17 $$1 - 984.T + 1.41e6T^{2}$$
19 $$1 - 1.14e3iT - 2.47e6T^{2}$$
23 $$1 - 578.T + 6.43e6T^{2}$$
29 $$1 + 5.86e3iT - 2.05e7T^{2}$$
31 $$1 + 9.82e3T + 2.86e7T^{2}$$
37 $$1 + 7.88e3iT - 6.93e7T^{2}$$
41 $$1 - 1.04e4T + 1.15e8T^{2}$$
43 $$1 - 1.90e4iT - 1.47e8T^{2}$$
47 $$1 - 2.97e3T + 2.29e8T^{2}$$
53 $$1 - 3.59e4iT - 4.18e8T^{2}$$
59 $$1 + 1.93e4iT - 7.14e8T^{2}$$
61 $$1 - 3.78e4iT - 8.44e8T^{2}$$
67 $$1 + 1.13e4iT - 1.35e9T^{2}$$
71 $$1 - 3.39e4T + 1.80e9T^{2}$$
73 $$1 - 2.27e4T + 2.07e9T^{2}$$
79 $$1 + 9.47e4T + 3.07e9T^{2}$$
83 $$1 + 3.47e4iT - 3.93e9T^{2}$$
89 $$1 + 4.20e4T + 5.58e9T^{2}$$
97 $$1 + 1.14e4T + 8.58e9T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}