# Properties

 Label 2-30e2-20.19-c2-0-27 Degree $2$ Conductor $900$ Sign $-0.447 - 0.894i$ Analytic cond. $24.5232$ Root an. cond. $4.95209$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 2i·2-s − 4·4-s − 8i·8-s − 10i·13-s + 16·16-s + 16i·17-s + 20·26-s + 40·29-s + 32i·32-s − 32·34-s + 70i·37-s + 80·41-s − 49·49-s + 40i·52-s + 56i·53-s + ⋯
 L(s)  = 1 + i·2-s − 4-s − i·8-s − 0.769i·13-s + 16-s + 0.941i·17-s + 0.769·26-s + 1.37·29-s + i·32-s − 0.941·34-s + 1.89i·37-s + 1.95·41-s − 0.999·49-s + 0.769i·52-s + 1.05i·53-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, 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: $$24.5232$$ Root analytic conductor: $$4.95209$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{900} (199, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 900,\ (\ :1),\ -0.447 - 0.894i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$1.454938921$$ $$L(\frac12)$$ $$\approx$$ $$1.454938921$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 - 2iT$$
3 $$1$$
5 $$1$$
good7 $$1 + 49T^{2}$$
11 $$1 - 121T^{2}$$
13 $$1 + 10iT - 169T^{2}$$
17 $$1 - 16iT - 289T^{2}$$
19 $$1 - 361T^{2}$$
23 $$1 + 529T^{2}$$
29 $$1 - 40T + 841T^{2}$$
31 $$1 - 961T^{2}$$
37 $$1 - 70iT - 1.36e3T^{2}$$
41 $$1 - 80T + 1.68e3T^{2}$$
43 $$1 + 1.84e3T^{2}$$
47 $$1 + 2.20e3T^{2}$$
53 $$1 - 56iT - 2.80e3T^{2}$$
59 $$1 - 3.48e3T^{2}$$
61 $$1 + 22T + 3.72e3T^{2}$$
67 $$1 + 4.48e3T^{2}$$
71 $$1 - 5.04e3T^{2}$$
73 $$1 - 110iT - 5.32e3T^{2}$$
79 $$1 - 6.24e3T^{2}$$
83 $$1 + 6.88e3T^{2}$$
89 $$1 - 160T + 7.92e3T^{2}$$
97 $$1 - 130iT - 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$