# Properties

 Label 2-90-5.4-c5-0-5 Degree $2$ Conductor $90$ Sign $0.178 - 0.983i$ Analytic cond. $14.4345$ Root an. cond. $3.79928$ Motivic weight $5$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 4i·2-s − 16·4-s + (55 + 10i)5-s + 4i·7-s − 64i·8-s + (−40 + 220i)10-s + 500·11-s − 288i·13-s − 16·14-s + 256·16-s + 1.51e3i·17-s + 1.34e3·19-s + (−880 − 160i)20-s + 2.00e3i·22-s + 4.10e3i·23-s + ⋯
 L(s)  = 1 + 0.707i·2-s − 0.5·4-s + (0.983 + 0.178i)5-s + 0.0308i·7-s − 0.353i·8-s + (−0.126 + 0.695i)10-s + 1.24·11-s − 0.472i·13-s − 0.0218·14-s + 0.250·16-s + 1.27i·17-s + 0.854·19-s + (−0.491 − 0.0894i)20-s + 0.880i·22-s + 1.61i·23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.178 - 0.983i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.178 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$90$$    =    $$2 \cdot 3^{2} \cdot 5$$ Sign: $0.178 - 0.983i$ Analytic conductor: $$14.4345$$ Root analytic conductor: $$3.79928$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: $\chi_{90} (19, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 90,\ (\ :5/2),\ 0.178 - 0.983i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$1.63276 + 1.36267i$$ $$L(\frac12)$$ $$\approx$$ $$1.63276 + 1.36267i$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 - 4iT$$
3 $$1$$
5 $$1 + (-55 - 10i)T$$
good7 $$1 - 4iT - 1.68e4T^{2}$$
11 $$1 - 500T + 1.61e5T^{2}$$
13 $$1 + 288iT - 3.71e5T^{2}$$
17 $$1 - 1.51e3iT - 1.41e6T^{2}$$
19 $$1 - 1.34e3T + 2.47e6T^{2}$$
23 $$1 - 4.10e3iT - 6.43e6T^{2}$$
29 $$1 + 2.64e3T + 2.05e7T^{2}$$
31 $$1 + 5.61e3T + 2.86e7T^{2}$$
37 $$1 - 7.28e3iT - 6.93e7T^{2}$$
41 $$1 - 1.89e4T + 1.15e8T^{2}$$
43 $$1 + 2.40e3iT - 1.47e8T^{2}$$
47 $$1 - 8.90e3iT - 2.29e8T^{2}$$
53 $$1 + 3.98e4iT - 4.18e8T^{2}$$
59 $$1 + 2.83e4T + 7.14e8T^{2}$$
61 $$1 - 1.82e4T + 8.44e8T^{2}$$
67 $$1 + 6.59e4iT - 1.35e9T^{2}$$
71 $$1 - 2.88e4T + 1.80e9T^{2}$$
73 $$1 + 3.08e4iT - 2.07e9T^{2}$$
79 $$1 + 6.02e4T + 3.07e9T^{2}$$
83 $$1 - 2.46e3iT - 3.93e9T^{2}$$
89 $$1 - 2.26e4T + 5.58e9T^{2}$$
97 $$1 - 3.69e4iT - 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$