# Properties

 Degree 2 Conductor $2^{6} \cdot 5$ Sign $1$ Motivic weight 5 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 0.755·3-s + 25·5-s − 172.·7-s − 242.·9-s − 391.·11-s − 149.·13-s + 18.8·15-s + 1.18e3·17-s − 685.·19-s − 130.·21-s + 996.·23-s + 625·25-s − 366.·27-s + 8.76e3·29-s + 9.52e3·31-s − 295.·33-s − 4.31e3·35-s − 1.02e4·37-s − 112.·39-s + 32.6·41-s − 1.03e4·43-s − 6.06e3·45-s + 1.69e4·47-s + 1.29e4·49-s + 897.·51-s + 2.22e4·53-s − 9.78e3·55-s + ⋯
 L(s)  = 1 + 0.0484·3-s + 0.447·5-s − 1.33·7-s − 0.997·9-s − 0.975·11-s − 0.244·13-s + 0.0216·15-s + 0.996·17-s − 0.435·19-s − 0.0644·21-s + 0.392·23-s + 0.200·25-s − 0.0968·27-s + 1.93·29-s + 1.78·31-s − 0.0472·33-s − 0.594·35-s − 1.22·37-s − 0.0118·39-s + 0.00303·41-s − 0.851·43-s − 0.446·45-s + 1.11·47-s + 0.768·49-s + 0.0483·51-s + 1.08·53-s − 0.436·55-s + ⋯

## Functional equation

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

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$320$$    =    $$2^{6} \cdot 5$$ $$\varepsilon$$ = $1$ motivic weight = $$5$$ character : $\chi_{320} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = $$0$$ Selberg data = $$(2,\ 320,\ (\ :5/2),\ 1)$$ $$L(3)$$ $$\approx$$ $$1.374828783$$ $$L(\frac12)$$ $$\approx$$ $$1.374828783$$ $$L(\frac{7}{2})$$ 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 - 25T$$
good3 $$1 - 0.755T + 243T^{2}$$
7 $$1 + 172.T + 1.68e4T^{2}$$
11 $$1 + 391.T + 1.61e5T^{2}$$
13 $$1 + 149.T + 3.71e5T^{2}$$
17 $$1 - 1.18e3T + 1.41e6T^{2}$$
19 $$1 + 685.T + 2.47e6T^{2}$$
23 $$1 - 996.T + 6.43e6T^{2}$$
29 $$1 - 8.76e3T + 2.05e7T^{2}$$
31 $$1 - 9.52e3T + 2.86e7T^{2}$$
37 $$1 + 1.02e4T + 6.93e7T^{2}$$
41 $$1 - 32.6T + 1.15e8T^{2}$$
43 $$1 + 1.03e4T + 1.47e8T^{2}$$
47 $$1 - 1.69e4T + 2.29e8T^{2}$$
53 $$1 - 2.22e4T + 4.18e8T^{2}$$
59 $$1 + 4.42e4T + 7.14e8T^{2}$$
61 $$1 - 2.17e4T + 8.44e8T^{2}$$
67 $$1 + 2.07e4T + 1.35e9T^{2}$$
71 $$1 - 1.53e4T + 1.80e9T^{2}$$
73 $$1 - 5.79e4T + 2.07e9T^{2}$$
79 $$1 + 6.24e4T + 3.07e9T^{2}$$
83 $$1 - 4.35e4T + 3.93e9T^{2}$$
89 $$1 - 6.62e4T + 5.58e9T^{2}$$
97 $$1 + 1.22e4T + 8.58e9T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}