# Properties

 Label 2-90-5.2-c2-0-3 Degree $2$ Conductor $90$ Sign $0.767 - 0.640i$ Analytic cond. $2.45232$ Root an. cond. $1.56598$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (1 + i)2-s + 2i·4-s + (3 − 4i)5-s + (8 + 8i)7-s + (−2 + 2i)8-s + (7 − i)10-s − 4·11-s + (−3 + 3i)13-s + 16i·14-s − 4·16-s + (−19 − 19i)17-s − 8i·19-s + (8 + 6i)20-s + (−4 − 4i)22-s + (20 − 20i)23-s + ⋯
 L(s)  = 1 + (0.5 + 0.5i)2-s + 0.5i·4-s + (0.600 − 0.800i)5-s + (1.14 + 1.14i)7-s + (−0.250 + 0.250i)8-s + (0.700 − 0.100i)10-s − 0.363·11-s + (−0.230 + 0.230i)13-s + 1.14i·14-s − 0.250·16-s + (−1.11 − 1.11i)17-s − 0.421i·19-s + (0.400 + 0.300i)20-s + (−0.181 − 0.181i)22-s + (0.869 − 0.869i)23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.767 - 0.640i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.767 - 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$90$$    =    $$2 \cdot 3^{2} \cdot 5$$ Sign: $0.767 - 0.640i$ Analytic conductor: $$2.45232$$ Root analytic conductor: $$1.56598$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{90} (37, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 90,\ (\ :1),\ 0.767 - 0.640i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$1.73252 + 0.627978i$$ $$L(\frac12)$$ $$\approx$$ $$1.73252 + 0.627978i$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-1 - i)T$$
3 $$1$$
5 $$1 + (-3 + 4i)T$$
good7 $$1 + (-8 - 8i)T + 49iT^{2}$$
11 $$1 + 4T + 121T^{2}$$
13 $$1 + (3 - 3i)T - 169iT^{2}$$
17 $$1 + (19 + 19i)T + 289iT^{2}$$
19 $$1 + 8iT - 361T^{2}$$
23 $$1 + (-20 + 20i)T - 529iT^{2}$$
29 $$1 - 38iT - 841T^{2}$$
31 $$1 + 44T + 961T^{2}$$
37 $$1 + (3 + 3i)T + 1.36e3iT^{2}$$
41 $$1 + 70T + 1.68e3T^{2}$$
43 $$1 + (-36 + 36i)T - 1.84e3iT^{2}$$
47 $$1 + 2.20e3iT^{2}$$
53 $$1 + (17 - 17i)T - 2.80e3iT^{2}$$
59 $$1 - 92iT - 3.48e3T^{2}$$
61 $$1 - 72T + 3.72e3T^{2}$$
67 $$1 + (-44 - 44i)T + 4.48e3iT^{2}$$
71 $$1 - 88T + 5.04e3T^{2}$$
73 $$1 + (-55 + 55i)T - 5.32e3iT^{2}$$
79 $$1 - 12iT - 6.24e3T^{2}$$
83 $$1 + (-24 + 24i)T - 6.88e3iT^{2}$$
89 $$1 + 26iT - 7.92e3T^{2}$$
97 $$1 + (57 + 57i)T + 9.40e3iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$