# Properties

 Degree 2 Conductor $2^{6} \cdot 5$ Sign $0.914 + 0.405i$ Motivic weight 5 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 19.1·3-s + (−55.2 − 8.66i)5-s + 121. i·7-s + 123.·9-s − 298i·11-s + 485.·13-s + (−1.05e3 − 165. i)15-s − 420. i·17-s − 2.57e3i·19-s + 2.32e3i·21-s + 717. i·23-s + (2.97e3 + 956. i)25-s − 2.29e3·27-s + 3.73e3i·29-s + 6.22e3·31-s + ⋯
 L(s)  = 1 + 1.22·3-s + (−0.987 − 0.154i)5-s + 0.937i·7-s + 0.506·9-s − 0.742i·11-s + 0.797·13-s + (−1.21 − 0.190i)15-s − 0.353i·17-s − 1.63i·19-s + 1.15i·21-s + 0.282i·23-s + (0.952 + 0.306i)25-s − 0.606·27-s + 0.824i·29-s + 1.16·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.405i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.914 + 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$320$$    =    $$2^{6} \cdot 5$$ $$\varepsilon$$ = $0.914 + 0.405i$ motivic weight = $$5$$ character : $\chi_{320} (289, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 320,\ (\ :5/2),\ 0.914 + 0.405i)$$ $$L(3)$$ $$\approx$$ $$2.675300987$$ $$L(\frac12)$$ $$\approx$$ $$2.675300987$$ $$L(\frac{7}{2})$$ 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 + (55.2 + 8.66i)T$$
good3 $$1 - 19.1T + 243T^{2}$$
7 $$1 - 121. iT - 1.68e4T^{2}$$
11 $$1 + 298iT - 1.61e5T^{2}$$
13 $$1 - 485.T + 3.71e5T^{2}$$
17 $$1 + 420. iT - 1.41e6T^{2}$$
19 $$1 + 2.57e3iT - 2.47e6T^{2}$$
23 $$1 - 717. iT - 6.43e6T^{2}$$
29 $$1 - 3.73e3iT - 2.05e7T^{2}$$
31 $$1 - 6.22e3T + 2.86e7T^{2}$$
37 $$1 - 5.72e3T + 6.93e7T^{2}$$
41 $$1 - 1.13e4T + 1.15e8T^{2}$$
43 $$1 - 1.91e4T + 1.47e8T^{2}$$
47 $$1 + 2.28e3iT - 2.29e8T^{2}$$
53 $$1 - 1.72e4T + 4.18e8T^{2}$$
59 $$1 - 1.63e4iT - 7.14e8T^{2}$$
61 $$1 + 4.87e4iT - 8.44e8T^{2}$$
67 $$1 - 8.39e3T + 1.35e9T^{2}$$
71 $$1 + 2.90e4T + 1.80e9T^{2}$$
73 $$1 + 420. iT - 2.07e9T^{2}$$
79 $$1 + 3.11e4T + 3.07e9T^{2}$$
83 $$1 - 1.06e5T + 3.93e9T^{2}$$
89 $$1 + 9.93e4T + 5.58e9T^{2}$$
97 $$1 + 1.80e5iT - 8.58e9T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}