# Properties

 Degree 2 Conductor $2^{6} \cdot 5$ Sign $0.987 + 0.156i$ Motivic weight 5 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 28.5·3-s + (5.81 + 55.5i)5-s + 92.3i·7-s + 571.·9-s − 541. i·11-s − 782.·13-s + (−165. − 1.58e3i)15-s + 1.55e3i·17-s − 484. i·19-s − 2.63e3i·21-s − 1.09e3i·23-s + (−3.05e3 + 646. i)25-s − 9.37e3·27-s − 597. i·29-s − 8.86e3·31-s + ⋯
 L(s)  = 1 − 1.83·3-s + (0.104 + 0.994i)5-s + 0.712i·7-s + 2.35·9-s − 1.34i·11-s − 1.28·13-s + (−0.190 − 1.82i)15-s + 1.30i·17-s − 0.307i·19-s − 1.30i·21-s − 0.432i·23-s + (−0.978 + 0.206i)25-s − 2.47·27-s − 0.131i·29-s − 1.65·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.156i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.987 + 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$320$$    =    $$2^{6} \cdot 5$$ $$\varepsilon$$ = $0.987 + 0.156i$ motivic weight = $$5$$ character : $\chi_{320} (289, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 320,\ (\ :5/2),\ 0.987 + 0.156i)$$ $$L(3)$$ $$\approx$$ $$0.5522652773$$ $$L(\frac12)$$ $$\approx$$ $$0.5522652773$$ $$L(\frac{7}{2})$$ 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 + (-5.81 - 55.5i)T$$
good3 $$1 + 28.5T + 243T^{2}$$
7 $$1 - 92.3iT - 1.68e4T^{2}$$
11 $$1 + 541. iT - 1.61e5T^{2}$$
13 $$1 + 782.T + 3.71e5T^{2}$$
17 $$1 - 1.55e3iT - 1.41e6T^{2}$$
19 $$1 + 484. iT - 2.47e6T^{2}$$
23 $$1 + 1.09e3iT - 6.43e6T^{2}$$
29 $$1 + 597. iT - 2.05e7T^{2}$$
31 $$1 + 8.86e3T + 2.86e7T^{2}$$
37 $$1 + 1.36e4T + 6.93e7T^{2}$$
41 $$1 - 1.45e4T + 1.15e8T^{2}$$
43 $$1 + 1.82e3T + 1.47e8T^{2}$$
47 $$1 + 1.79e4iT - 2.29e8T^{2}$$
53 $$1 - 1.65e3T + 4.18e8T^{2}$$
59 $$1 - 1.44e4iT - 7.14e8T^{2}$$
61 $$1 + 2.45e4iT - 8.44e8T^{2}$$
67 $$1 - 2.68e4T + 1.35e9T^{2}$$
71 $$1 + 4.80e4T + 1.80e9T^{2}$$
73 $$1 + 6.06e4iT - 2.07e9T^{2}$$
79 $$1 - 2.81e4T + 3.07e9T^{2}$$
83 $$1 - 2.45e4T + 3.93e9T^{2}$$
89 $$1 - 1.17e5T + 5.58e9T^{2}$$
97 $$1 + 3.77e4iT - 8.58e9T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}