# Properties

 Degree 2 Conductor $2^{2} \cdot 5$ Sign $1$ Motivic weight 4 Primitive yes Self-dual yes Analytic rank 0

# Origins

## Dirichlet series

 L(s)  = 1 + 4·2-s − 2·3-s + 16·4-s + 25·5-s − 8·6-s − 82·7-s + 64·8-s − 77·9-s + 100·10-s − 32·12-s − 328·14-s − 50·15-s + 256·16-s − 308·18-s + 400·20-s + 164·21-s + 878·23-s − 128·24-s + 625·25-s + 316·27-s − 1.31e3·28-s − 1.19e3·29-s − 200·30-s + 1.02e3·32-s − 2.05e3·35-s − 1.23e3·36-s + 1.60e3·40-s + ⋯
 L(s)  = 1 + 2-s − 2/9·3-s + 4-s + 5-s − 2/9·6-s − 1.67·7-s + 8-s − 0.950·9-s + 10-s − 2/9·12-s − 1.67·14-s − 2/9·15-s + 16-s − 0.950·18-s + 20-s + 0.371·21-s + 1.65·23-s − 2/9·24-s + 25-s + 0.433·27-s − 1.67·28-s − 1.42·29-s − 2/9·30-s + 32-s − 1.67·35-s − 0.950·36-s + 40-s + ⋯

## Functional equation

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

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$20$$    =    $$2^{2} \cdot 5$$ $$\varepsilon$$ = $1$ motivic weight = $$4$$ character : $\chi_{20} (19, \cdot )$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 20,\ (\ :2),\ 1)$ $L(\frac{5}{2})$ $\approx$ $1.97006$ $L(\frac12)$ $\approx$ $1.97006$ $L(3)$ 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 - p^{2} T$$
5 $$1 - p^{2} T$$
good3 $$1 + 2 T + p^{4} T^{2}$$
7 $$1 + 82 T + p^{4} T^{2}$$
11 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
13 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
17 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
19 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
23 $$1 - 878 T + p^{4} T^{2}$$
29 $$1 + 1198 T + p^{4} T^{2}$$
31 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
37 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
41 $$1 - 482 T + p^{4} T^{2}$$
43 $$1 - 2078 T + p^{4} T^{2}$$
47 $$1 + 4402 T + p^{4} T^{2}$$
53 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
59 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
61 $$1 + 4078 T + p^{4} T^{2}$$
67 $$1 - 4478 T + p^{4} T^{2}$$
71 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
73 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
79 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
83 $$1 + 8002 T + p^{4} T^{2}$$
89 $$1 - 4322 T + p^{4} T^{2}$$
97 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}