# Properties

 Label 2-30e2-5.4-c5-0-19 Degree $2$ Conductor $900$ Sign $0.447 + 0.894i$ Analytic cond. $144.345$ Root an. cond. $12.0143$ Motivic weight $5$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 88i·7-s − 540·11-s − 418i·13-s + 594i·17-s − 836·19-s + 4.10e3i·23-s − 594·29-s + 4.25e3·31-s + 298i·37-s − 1.72e4·41-s − 1.21e4i·43-s − 1.29e3i·47-s + 9.06e3·49-s − 1.94e4i·53-s − 7.66e3·59-s + ⋯
 L(s)  = 1 + 0.678i·7-s − 1.34·11-s − 0.685i·13-s + 0.498i·17-s − 0.531·19-s + 1.61i·23-s − 0.131·29-s + 0.795·31-s + 0.0357i·37-s − 1.60·41-s − 0.997i·43-s − 0.0855i·47-s + 0.539·49-s − 0.953i·53-s − 0.286·59-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$900$$    =    $$2^{2} \cdot 3^{2} \cdot 5^{2}$$ Sign: $0.447 + 0.894i$ Analytic conductor: $$144.345$$ Root analytic conductor: $$12.0143$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: $\chi_{900} (649, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 900,\ (\ :5/2),\ 0.447 + 0.894i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$1.031853890$$ $$L(\frac12)$$ $$\approx$$ $$1.031853890$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
5 $$1$$
good7 $$1 - 88iT - 1.68e4T^{2}$$
11 $$1 + 540T + 1.61e5T^{2}$$
13 $$1 + 418iT - 3.71e5T^{2}$$
17 $$1 - 594iT - 1.41e6T^{2}$$
19 $$1 + 836T + 2.47e6T^{2}$$
23 $$1 - 4.10e3iT - 6.43e6T^{2}$$
29 $$1 + 594T + 2.05e7T^{2}$$
31 $$1 - 4.25e3T + 2.86e7T^{2}$$
37 $$1 - 298iT - 6.93e7T^{2}$$
41 $$1 + 1.72e4T + 1.15e8T^{2}$$
43 $$1 + 1.21e4iT - 1.47e8T^{2}$$
47 $$1 + 1.29e3iT - 2.29e8T^{2}$$
53 $$1 + 1.94e4iT - 4.18e8T^{2}$$
59 $$1 + 7.66e3T + 7.14e8T^{2}$$
61 $$1 + 3.47e4T + 8.44e8T^{2}$$
67 $$1 + 2.18e4iT - 1.35e9T^{2}$$
71 $$1 - 4.68e4T + 1.80e9T^{2}$$
73 $$1 - 6.75e4iT - 2.07e9T^{2}$$
79 $$1 - 7.69e4T + 3.07e9T^{2}$$
83 $$1 + 6.77e4iT - 3.93e9T^{2}$$
89 $$1 - 2.97e4T + 5.58e9T^{2}$$
97 $$1 - 1.22e5iT - 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$