# 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 + 6.32·3-s − 25·5-s − 44.2·7-s − 203·9-s − 720.·11-s + 146·13-s − 158.·15-s − 702·17-s + 2.73e3·19-s − 280·21-s + 4.09e3·23-s + 625·25-s − 2.82e3·27-s + 4.01e3·29-s − 4.56e3·31-s − 4.56e3·33-s + 1.10e3·35-s + 1.47e4·37-s + 923.·39-s − 4.35e3·41-s + 1.24e4·43-s + 5.07e3·45-s − 6.01e3·47-s − 1.48e4·49-s − 4.43e3·51-s + 1.81e4·53-s + 1.80e4·55-s + ⋯
 L(s)  = 1 + 0.405·3-s − 0.447·5-s − 0.341·7-s − 0.835·9-s − 1.79·11-s + 0.239·13-s − 0.181·15-s − 0.589·17-s + 1.73·19-s − 0.138·21-s + 1.61·23-s + 0.200·25-s − 0.744·27-s + 0.885·29-s − 0.853·31-s − 0.728·33-s + 0.152·35-s + 1.77·37-s + 0.0972·39-s − 0.404·41-s + 1.02·43-s + 0.373·45-s − 0.397·47-s − 0.883·49-s − 0.239·51-s + 0.887·53-s + 0.803·55-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 = $$(2,\ 320,\ (\ :5/2),\ 1)$$ $$L(3)$$ $$\approx$$ $$1.509982416$$ $$L(\frac12)$$ $$\approx$$ $$1.509982416$$ $$L(\frac{7}{2})$$ 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 - 6.32T + 243T^{2}$$
7 $$1 + 44.2T + 1.68e4T^{2}$$
11 $$1 + 720.T + 1.61e5T^{2}$$
13 $$1 - 146T + 3.71e5T^{2}$$
17 $$1 + 702T + 1.41e6T^{2}$$
19 $$1 - 2.73e3T + 2.47e6T^{2}$$
23 $$1 - 4.09e3T + 6.43e6T^{2}$$
29 $$1 - 4.01e3T + 2.05e7T^{2}$$
31 $$1 + 4.56e3T + 2.86e7T^{2}$$
37 $$1 - 1.47e4T + 6.93e7T^{2}$$
41 $$1 + 4.35e3T + 1.15e8T^{2}$$
43 $$1 - 1.24e4T + 1.47e8T^{2}$$
47 $$1 + 6.01e3T + 2.29e8T^{2}$$
53 $$1 - 1.81e4T + 4.18e8T^{2}$$
59 $$1 + 1.97e4T + 7.14e8T^{2}$$
61 $$1 - 4.21e4T + 8.44e8T^{2}$$
67 $$1 - 1.61e4T + 1.35e9T^{2}$$
71 $$1 + 4.54e4T + 1.80e9T^{2}$$
73 $$1 - 2.62e4T + 2.07e9T^{2}$$
79 $$1 - 8.67e3T + 3.07e9T^{2}$$
83 $$1 + 9.87e4T + 3.93e9T^{2}$$
89 $$1 - 3.05e4T + 5.58e9T^{2}$$
97 $$1 - 6.68e4T + 8.58e9T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}