# Properties

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

# Origins

## Dirichlet series

 L(s)  = 1 + 16·2-s + 174·3-s + 256·4-s − 625·5-s + 2.78e3·6-s + 4.65e3·7-s + 4.09e3·8-s + 1.05e4·9-s − 1.00e4·10-s + 2.89e4·11-s + 4.45e4·12-s − 1.64e5·13-s + 7.45e4·14-s − 1.08e5·15-s + 6.55e4·16-s − 5.94e5·17-s + 1.69e5·18-s − 2.95e5·19-s − 1.60e5·20-s + 8.10e5·21-s + 4.63e5·22-s + 2.54e6·23-s + 7.12e5·24-s + 3.90e5·25-s − 2.63e6·26-s − 1.58e6·27-s + 1.19e6·28-s + ⋯
 L(s)  = 1 + 0.707·2-s + 1.24·3-s + 1/2·4-s − 0.447·5-s + 0.876·6-s + 0.733·7-s + 0.353·8-s + 0.538·9-s − 0.316·10-s + 0.597·11-s + 0.620·12-s − 1.59·13-s + 0.518·14-s − 0.554·15-s + 1/4·16-s − 1.72·17-s + 0.380·18-s − 0.520·19-s − 0.223·20-s + 0.909·21-s + 0.422·22-s + 1.89·23-s + 0.438·24-s + 1/5·25-s − 1.12·26-s − 0.572·27-s + 0.366·28-s + ⋯

## Functional equation

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

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$10$$    =    $$2 \cdot 5$$ $$\varepsilon$$ = $1$ motivic weight = $$9$$ character : $\chi_{10} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 10,\ (\ :9/2),\ 1)$ $L(5)$ $\approx$ $2.99287$ $L(\frac12)$ $\approx$ $2.99287$ $L(\frac{11}{2})$ 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 - p^{4} T$$
5 $$1 + p^{4} T$$
good3 $$1 - 58 p T + p^{9} T^{2}$$
7 $$1 - 4658 T + p^{9} T^{2}$$
11 $$1 - 28992 T + p^{9} T^{2}$$
13 $$1 + 164446 T + p^{9} T^{2}$$
17 $$1 + 594822 T + p^{9} T^{2}$$
19 $$1 + 295780 T + p^{9} T^{2}$$
23 $$1 - 2544534 T + p^{9} T^{2}$$
29 $$1 + 3722970 T + p^{9} T^{2}$$
31 $$1 - 2335772 T + p^{9} T^{2}$$
37 $$1 - 10840418 T + p^{9} T^{2}$$
41 $$1 - 21593862 T + p^{9} T^{2}$$
43 $$1 - 10832294 T + p^{9} T^{2}$$
47 $$1 - 5172138 T + p^{9} T^{2}$$
53 $$1 - 98179674 T + p^{9} T^{2}$$
59 $$1 - 16162860 T + p^{9} T^{2}$$
61 $$1 + 43928158 T + p^{9} T^{2}$$
67 $$1 + 81557422 T + p^{9} T^{2}$$
71 $$1 - 161307732 T + p^{9} T^{2}$$
73 $$1 + 247147966 T + p^{9} T^{2}$$
79 $$1 + 583345720 T + p^{9} T^{2}$$
83 $$1 + 14571786 T + p^{9} T^{2}$$
89 $$1 - 470133690 T + p^{9} T^{2}$$
97 $$1 + 117838462 T + p^{9} T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}