# Properties

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

# Origins

## Dirichlet series

 L(s)  = 1 + 22·3-s − 25·5-s + 218·7-s + 241·9-s − 480·11-s − 622·13-s − 550·15-s + 186·17-s − 1.20e3·19-s + 4.79e3·21-s − 3.18e3·23-s + 625·25-s − 44·27-s + 5.52e3·29-s + 9.35e3·31-s − 1.05e4·33-s − 5.45e3·35-s + 5.61e3·37-s − 1.36e4·39-s − 1.43e4·41-s − 370·43-s − 6.02e3·45-s + 1.61e4·47-s + 3.07e4·49-s + 4.09e3·51-s − 4.37e3·53-s + 1.20e4·55-s + ⋯
 L(s)  = 1 + 1.41·3-s − 0.447·5-s + 1.68·7-s + 0.991·9-s − 1.19·11-s − 1.02·13-s − 0.631·15-s + 0.156·17-s − 0.765·19-s + 2.37·21-s − 1.25·23-s + 1/5·25-s − 0.0116·27-s + 1.22·29-s + 1.74·31-s − 1.68·33-s − 0.752·35-s + 0.674·37-s − 1.44·39-s − 1.33·41-s − 0.0305·43-s − 0.443·45-s + 1.06·47-s + 1.82·49-s + 0.220·51-s − 0.213·53-s + 0.534·55-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$20$$    =    $$2^{2} \cdot 5$$ $$\varepsilon$$ = $1$ motivic weight = $$5$$ character : $\chi_{20} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 20,\ (\ :5/2),\ 1)$ $L(3)$ $\approx$ $1.99658$ $L(\frac12)$ $\approx$ $1.99658$ $L(\frac{7}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;5\}$, $$F_p$$ is a polynomial of degree 2. If $p \in \{2,\;5\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 $$1$$
5 $$1 + p^{2} T$$
good3 $$1 - 22 T + p^{5} T^{2}$$
7 $$1 - 218 T + p^{5} T^{2}$$
11 $$1 + 480 T + p^{5} T^{2}$$
13 $$1 + 622 T + p^{5} T^{2}$$
17 $$1 - 186 T + p^{5} T^{2}$$
19 $$1 + 1204 T + p^{5} T^{2}$$
23 $$1 + 3186 T + p^{5} T^{2}$$
29 $$1 - 5526 T + p^{5} T^{2}$$
31 $$1 - 9356 T + p^{5} T^{2}$$
37 $$1 - 5618 T + p^{5} T^{2}$$
41 $$1 + 14394 T + p^{5} T^{2}$$
43 $$1 + 370 T + p^{5} T^{2}$$
47 $$1 - 16146 T + p^{5} T^{2}$$
53 $$1 + 4374 T + p^{5} T^{2}$$
59 $$1 + 11748 T + p^{5} T^{2}$$
61 $$1 - 13202 T + p^{5} T^{2}$$
67 $$1 + 11542 T + p^{5} T^{2}$$
71 $$1 + 29532 T + p^{5} T^{2}$$
73 $$1 - 33698 T + p^{5} T^{2}$$
79 $$1 - 31208 T + p^{5} T^{2}$$
83 $$1 + 38466 T + p^{5} T^{2}$$
89 $$1 - 119514 T + p^{5} T^{2}$$
97 $$1 - 94658 T + p^{5} T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}