# Properties

 Degree 2 Conductor $2 \cdot 3^{2} \cdot 5^{2}$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 2-s + 4-s + 4·7-s − 8-s − 2·13-s − 4·14-s + 16-s + 6·17-s − 4·19-s + 2·26-s + 4·28-s + 6·29-s + 8·31-s − 32-s − 6·34-s − 2·37-s + 4·38-s + 6·41-s + 4·43-s + 9·49-s − 2·52-s − 6·53-s − 4·56-s − 6·58-s − 10·61-s − 8·62-s + 64-s + ⋯
 L(s)  = 1 − 0.707·2-s + 1/2·4-s + 1.51·7-s − 0.353·8-s − 0.554·13-s − 1.06·14-s + 1/4·16-s + 1.45·17-s − 0.917·19-s + 0.392·26-s + 0.755·28-s + 1.11·29-s + 1.43·31-s − 0.176·32-s − 1.02·34-s − 0.328·37-s + 0.648·38-s + 0.937·41-s + 0.609·43-s + 9/7·49-s − 0.277·52-s − 0.824·53-s − 0.534·56-s − 0.787·58-s − 1.28·61-s − 1.01·62-s + 1/8·64-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$450$$    =    $$2 \cdot 3^{2} \cdot 5^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{450} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = $$0$$ Selberg data = $$(2,\ 450,\ (\ :1/2),\ 1)$$ $$L(1)$$ $$\approx$$ $$1.19638$$ $$L(\frac12)$$ $$\approx$$ $$1.19638$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;5\}$,$F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 + T$$
3 $$1$$
5 $$1$$
good7 $$1 - 4 T + p T^{2}$$
11 $$1 + p T^{2}$$
13 $$1 + 2 T + p T^{2}$$
17 $$1 - 6 T + p T^{2}$$
19 $$1 + 4 T + p T^{2}$$
23 $$1 + p T^{2}$$
29 $$1 - 6 T + p T^{2}$$
31 $$1 - 8 T + p T^{2}$$
37 $$1 + 2 T + p T^{2}$$
41 $$1 - 6 T + p T^{2}$$
43 $$1 - 4 T + p T^{2}$$
47 $$1 + p T^{2}$$
53 $$1 + 6 T + p T^{2}$$
59 $$1 + p T^{2}$$
61 $$1 + 10 T + p T^{2}$$
67 $$1 - 4 T + p T^{2}$$
71 $$1 + p T^{2}$$
73 $$1 + 2 T + p T^{2}$$
79 $$1 - 8 T + p T^{2}$$
83 $$1 - 12 T + p T^{2}$$
89 $$1 + 18 T + p T^{2}$$
97 $$1 + 2 T + p T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}