# Properties

 Label 2-450-1.1-c5-0-33 Degree $2$ Conductor $450$ Sign $-1$ Analytic cond. $72.1727$ Root an. cond. $8.49545$ Motivic weight $5$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

# Origins

## Dirichlet series

 L(s)  = 1 + 4·2-s + 16·4-s − 47·7-s + 64·8-s − 222·11-s − 101·13-s − 188·14-s + 256·16-s + 162·17-s + 1.68e3·19-s − 888·22-s + 306·23-s − 404·26-s − 752·28-s − 7.89e3·29-s − 8.59e3·31-s + 1.02e3·32-s + 648·34-s − 8.64e3·37-s + 6.74e3·38-s + 1.81e4·41-s − 1.43e4·43-s − 3.55e3·44-s + 1.22e3·46-s − 1.09e3·47-s − 1.45e4·49-s − 1.61e3·52-s + ⋯
 L(s)  = 1 + 0.707·2-s + 1/2·4-s − 0.362·7-s + 0.353·8-s − 0.553·11-s − 0.165·13-s − 0.256·14-s + 1/4·16-s + 0.135·17-s + 1.07·19-s − 0.391·22-s + 0.120·23-s − 0.117·26-s − 0.181·28-s − 1.74·29-s − 1.60·31-s + 0.176·32-s + 0.0961·34-s − 1.03·37-s + 0.757·38-s + 1.68·41-s − 1.18·43-s − 0.276·44-s + 0.0852·46-s − 0.0725·47-s − 0.868·49-s − 0.0828·52-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$450$$    =    $$2 \cdot 3^{2} \cdot 5^{2}$$ Sign: $-1$ Analytic conductor: $$72.1727$$ Root analytic conductor: $$8.49545$$ Motivic weight: $$5$$ Rational: yes Arithmetic: yes Character: $\chi_{450} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 450,\ (\ :5/2),\ -1)$$

## Particular Values

 $$L(3)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 - p^{2} T$$
3 $$1$$
5 $$1$$
good7 $$1 + 47 T + p^{5} T^{2}$$
11 $$1 + 222 T + p^{5} T^{2}$$
13 $$1 + 101 T + p^{5} T^{2}$$
17 $$1 - 162 T + p^{5} T^{2}$$
19 $$1 - 1685 T + p^{5} T^{2}$$
23 $$1 - 306 T + p^{5} T^{2}$$
29 $$1 + 7890 T + p^{5} T^{2}$$
31 $$1 + 8593 T + p^{5} T^{2}$$
37 $$1 + 8642 T + p^{5} T^{2}$$
41 $$1 - 18168 T + p^{5} T^{2}$$
43 $$1 + 14351 T + p^{5} T^{2}$$
47 $$1 + 1098 T + p^{5} T^{2}$$
53 $$1 - 17916 T + p^{5} T^{2}$$
59 $$1 + 17610 T + p^{5} T^{2}$$
61 $$1 + 21853 T + p^{5} T^{2}$$
67 $$1 + 107 T + p^{5} T^{2}$$
71 $$1 - 40728 T + p^{5} T^{2}$$
73 $$1 + 34706 T + p^{5} T^{2}$$
79 $$1 + 69160 T + p^{5} T^{2}$$
83 $$1 + 108534 T + p^{5} T^{2}$$
89 $$1 + 35040 T + p^{5} T^{2}$$
97 $$1 - 823 T + p^{5} T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$