# Properties

 Label 2-150-1.1-c5-0-8 Degree $2$ Conductor $150$ Sign $1$ Analytic cond. $24.0575$ Root an. cond. $4.90485$ Motivic weight $5$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 + 4·2-s + 9·3-s + 16·4-s + 36·6-s + 47·7-s + 64·8-s + 81·9-s + 222·11-s + 144·12-s + 101·13-s + 188·14-s + 256·16-s + 162·17-s + 324·18-s + 1.68e3·19-s + 423·21-s + 888·22-s + 306·23-s + 576·24-s + 404·26-s + 729·27-s + 752·28-s + 7.89e3·29-s − 8.59e3·31-s + 1.02e3·32-s + 1.99e3·33-s + 648·34-s + ⋯
 L(s)  = 1 + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.362·7-s + 0.353·8-s + 1/3·9-s + 0.553·11-s + 0.288·12-s + 0.165·13-s + 0.256·14-s + 1/4·16-s + 0.135·17-s + 0.235·18-s + 1.07·19-s + 0.209·21-s + 0.391·22-s + 0.120·23-s + 0.204·24-s + 0.117·26-s + 0.192·27-s + 0.181·28-s + 1.74·29-s − 1.60·31-s + 0.176·32-s + 0.319·33-s + 0.0961·34-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$150$$    =    $$2 \cdot 3 \cdot 5^{2}$$ Sign: $1$ Analytic conductor: $$24.0575$$ Root analytic conductor: $$4.90485$$ Motivic weight: $$5$$ Rational: yes Arithmetic: yes Character: $\chi_{150} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 150,\ (\ :5/2),\ 1)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$4.105786123$$ $$L(\frac12)$$ $$\approx$$ $$4.105786123$$ $$L(\frac{7}{2})$$ 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 - p^{2} T$$
5 $$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}$$