# Properties

 Degree 2 Conductor $2^{6} \cdot 5$ Sign $0.410 - 0.911i$ Motivic weight 5 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 4.76·3-s + (−43.2 − 35.3i)5-s − 82.4i·7-s − 220.·9-s − 109. i·11-s − 621.·13-s + (−206. − 168. i)15-s + 503. i·17-s + 853. i·19-s − 392. i·21-s − 1.42e3i·23-s + (621. + 3.06e3i)25-s − 2.20e3·27-s − 275. i·29-s + 6.20e3·31-s + ⋯
 L(s)  = 1 + 0.305·3-s + (−0.774 − 0.632i)5-s − 0.635i·7-s − 0.906·9-s − 0.272i·11-s − 1.02·13-s + (−0.236 − 0.193i)15-s + 0.422i·17-s + 0.542i·19-s − 0.194i·21-s − 0.561i·23-s + (0.198 + 0.980i)25-s − 0.583·27-s − 0.0607i·29-s + 1.16·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.410 - 0.911i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.410 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$320$$    =    $$2^{6} \cdot 5$$ $$\varepsilon$$ = $0.410 - 0.911i$ motivic weight = $$5$$ character : $\chi_{320} (289, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 320,\ (\ :5/2),\ 0.410 - 0.911i)$$ $$L(3)$$ $$\approx$$ $$0.8317471145$$ $$L(\frac12)$$ $$\approx$$ $$0.8317471145$$ $$L(\frac{7}{2})$$ 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 + (43.2 + 35.3i)T$$
good3 $$1 - 4.76T + 243T^{2}$$
7 $$1 + 82.4iT - 1.68e4T^{2}$$
11 $$1 + 109. iT - 1.61e5T^{2}$$
13 $$1 + 621.T + 3.71e5T^{2}$$
17 $$1 - 503. iT - 1.41e6T^{2}$$
19 $$1 - 853. iT - 2.47e6T^{2}$$
23 $$1 + 1.42e3iT - 6.43e6T^{2}$$
29 $$1 + 275. iT - 2.05e7T^{2}$$
31 $$1 - 6.20e3T + 2.86e7T^{2}$$
37 $$1 - 3.66e3T + 6.93e7T^{2}$$
41 $$1 + 1.26e3T + 1.15e8T^{2}$$
43 $$1 - 7.49e3T + 1.47e8T^{2}$$
47 $$1 - 1.58e4iT - 2.29e8T^{2}$$
53 $$1 - 1.00e4T + 4.18e8T^{2}$$
59 $$1 - 2.79e4iT - 7.14e8T^{2}$$
61 $$1 + 3.99e4iT - 8.44e8T^{2}$$
67 $$1 + 6.65e4T + 1.35e9T^{2}$$
71 $$1 - 3.27e4T + 1.80e9T^{2}$$
73 $$1 - 7.15e4iT - 2.07e9T^{2}$$
79 $$1 - 2.07e4T + 3.07e9T^{2}$$
83 $$1 + 1.08e5T + 3.93e9T^{2}$$
89 $$1 - 8.48e4T + 5.58e9T^{2}$$
97 $$1 - 3.83e4iT - 8.58e9T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}