# 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 + 55.9·5-s − 245. i·7-s − 243·9-s + 802i·11-s − 245.·13-s + 1.91e3i·19-s + 3.33e3i·23-s + 3.12e3·25-s − 1.37e4i·35-s + 1.58e4·37-s + 1.52e4·41-s − 1.35e4·45-s − 1.53e4i·47-s − 4.36e4·49-s + 1.68e4·53-s + ⋯
 L(s)  = 1 + 0.999·5-s − 1.89i·7-s − 9-s + 1.99i·11-s − 0.403·13-s + 1.21i·19-s + 1.31i·23-s + 25-s − 1.89i·35-s + 1.90·37-s + 1.41·41-s − 0.999·45-s − 1.01i·47-s − 2.59·49-s + 0.823·53-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 : $\chi_{320} (289, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 320,\ (\ :5/2),\ 0.707 - 0.707i)$$ $$L(3)$$ $$\approx$$ $$2.006085520$$ $$L(\frac12)$$ $$\approx$$ $$2.006085520$$ $$L(\frac{7}{2})$$ 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 - 55.9T$$
good3 $$1 + 243T^{2}$$
7 $$1 + 245. iT - 1.68e4T^{2}$$
11 $$1 - 802iT - 1.61e5T^{2}$$
13 $$1 + 245.T + 3.71e5T^{2}$$
17 $$1 - 1.41e6T^{2}$$
19 $$1 - 1.91e3iT - 2.47e6T^{2}$$
23 $$1 - 3.33e3iT - 6.43e6T^{2}$$
29 $$1 - 2.05e7T^{2}$$
31 $$1 + 2.86e7T^{2}$$
37 $$1 - 1.58e4T + 6.93e7T^{2}$$
41 $$1 - 1.52e4T + 1.15e8T^{2}$$
43 $$1 + 1.47e8T^{2}$$
47 $$1 + 1.53e4iT - 2.29e8T^{2}$$
53 $$1 - 1.68e4T + 4.18e8T^{2}$$
59 $$1 - 2.79e4iT - 7.14e8T^{2}$$
61 $$1 - 8.44e8T^{2}$$
67 $$1 + 1.35e9T^{2}$$
71 $$1 + 1.80e9T^{2}$$
73 $$1 - 2.07e9T^{2}$$
79 $$1 + 3.07e9T^{2}$$
83 $$1 + 3.93e9T^{2}$$
89 $$1 - 1.28e5T + 5.58e9T^{2}$$
97 $$1 - 8.58e9T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}