# Properties

 Degree 2 Conductor $2^{2} \cdot 5$ Sign $1$ Motivic weight 9 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 272.·3-s − 625·5-s − 1.00e4·7-s + 5.44e4·9-s + 4.70e4·11-s + 9.36e3·13-s + 1.70e5·15-s + 1.08e5·17-s − 6.65e5·19-s + 2.72e6·21-s + 5.76e5·23-s + 3.90e5·25-s − 9.45e6·27-s − 2.61e6·29-s + 3.87e6·31-s − 1.28e7·33-s + 6.25e6·35-s + 1.41e7·37-s − 2.54e6·39-s + 4.62e6·41-s + 8.31e6·43-s − 3.40e7·45-s + 2.51e7·47-s + 5.96e7·49-s − 2.95e7·51-s + 3.49e7·53-s − 2.94e7·55-s + ⋯
 L(s)  = 1 − 1.94·3-s − 0.447·5-s − 1.57·7-s + 2.76·9-s + 0.969·11-s + 0.0909·13-s + 0.867·15-s + 0.315·17-s − 1.17·19-s + 3.05·21-s + 0.429·23-s + 0.200·25-s − 3.42·27-s − 0.685·29-s + 0.754·31-s − 1.88·33-s + 0.704·35-s + 1.24·37-s − 0.176·39-s + 0.255·41-s + 0.370·43-s − 1.23·45-s + 0.753·47-s + 1.47·49-s − 0.611·51-s + 0.608·53-s − 0.433·55-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$20$$    =    $$2^{2} \cdot 5$$ $$\varepsilon$$ = $1$ motivic weight = $$9$$ character : $\chi_{20} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = $$0$$ Selberg data = $$(2,\ 20,\ (\ :9/2),\ 1)$$ $$L(5)$$ $$\approx$$ $$0.558347$$ $$L(\frac12)$$ $$\approx$$ $$0.558347$$ $$L(\frac{11}{2})$$ 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 + 625T$$
good3 $$1 + 272.T + 1.96e4T^{2}$$
7 $$1 + 1.00e4T + 4.03e7T^{2}$$
11 $$1 - 4.70e4T + 2.35e9T^{2}$$
13 $$1 - 9.36e3T + 1.06e10T^{2}$$
17 $$1 - 1.08e5T + 1.18e11T^{2}$$
19 $$1 + 6.65e5T + 3.22e11T^{2}$$
23 $$1 - 5.76e5T + 1.80e12T^{2}$$
29 $$1 + 2.61e6T + 1.45e13T^{2}$$
31 $$1 - 3.87e6T + 2.64e13T^{2}$$
37 $$1 - 1.41e7T + 1.29e14T^{2}$$
41 $$1 - 4.62e6T + 3.27e14T^{2}$$
43 $$1 - 8.31e6T + 5.02e14T^{2}$$
47 $$1 - 2.51e7T + 1.11e15T^{2}$$
53 $$1 - 3.49e7T + 3.29e15T^{2}$$
59 $$1 - 6.71e6T + 8.66e15T^{2}$$
61 $$1 + 4.75e6T + 1.16e16T^{2}$$
67 $$1 + 1.38e8T + 2.72e16T^{2}$$
71 $$1 - 3.54e8T + 4.58e16T^{2}$$
73 $$1 - 2.41e8T + 5.88e16T^{2}$$
79 $$1 - 2.61e8T + 1.19e17T^{2}$$
83 $$1 + 6.55e8T + 1.86e17T^{2}$$
89 $$1 + 1.00e9T + 3.50e17T^{2}$$
97 $$1 - 1.24e9T + 7.60e17T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}