# Properties

 Label 2-30e2-4.3-c2-0-53 Degree $2$ Conductor $900$ Sign $0.875 - 0.484i$ 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 + (0.5 + 1.93i)2-s + (−3.50 + 1.93i)4-s + (−5.50 − 5.80i)8-s + (8.50 − 13.5i)16-s + 14·17-s − 30.9i·19-s − 30.9i·23-s + 61.9i·31-s + (30.5 + 9.68i)32-s + (7 + 27.1i)34-s + (60.0 − 15.4i)38-s + (60.0 − 15.4i)46-s − 92.9i·47-s + 49·49-s + 86·53-s + ⋯
 L(s)  = 1 + (0.250 + 0.968i)2-s + (−0.875 + 0.484i)4-s + (−0.687 − 0.726i)8-s + (0.531 − 0.847i)16-s + 0.823·17-s − 1.63i·19-s − 1.34i·23-s + 1.99i·31-s + (0.953 + 0.302i)32-s + (0.205 + 0.797i)34-s + (1.57 − 0.407i)38-s + (1.30 − 0.336i)46-s − 1.97i·47-s + 0.999·49-s + 1.62·53-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 - 0.484i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.875 - 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$900$$    =    $$2^{2} \cdot 3^{2} \cdot 5^{2}$$ Sign: $0.875 - 0.484i$ Analytic conductor: $$24.5232$$ Root analytic conductor: $$4.95209$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{900} (451, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 900,\ (\ :1),\ 0.875 - 0.484i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$1.748149278$$ $$L(\frac12)$$ $$\approx$$ $$1.748149278$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-0.5 - 1.93i)T$$
3 $$1$$
5 $$1$$
good7 $$1 - 49T^{2}$$
11 $$1 - 121T^{2}$$
13 $$1 + 169T^{2}$$
17 $$1 - 14T + 289T^{2}$$
19 $$1 + 30.9iT - 361T^{2}$$
23 $$1 + 30.9iT - 529T^{2}$$
29 $$1 + 841T^{2}$$
31 $$1 - 61.9iT - 961T^{2}$$
37 $$1 + 1.36e3T^{2}$$
41 $$1 + 1.68e3T^{2}$$
43 $$1 - 1.84e3T^{2}$$
47 $$1 + 92.9iT - 2.20e3T^{2}$$
53 $$1 - 86T + 2.80e3T^{2}$$
59 $$1 - 3.48e3T^{2}$$
61 $$1 - 118T + 3.72e3T^{2}$$
67 $$1 - 4.48e3T^{2}$$
71 $$1 - 5.04e3T^{2}$$
73 $$1 + 5.32e3T^{2}$$
79 $$1 - 123. iT - 6.24e3T^{2}$$
83 $$1 + 61.9iT - 6.88e3T^{2}$$
89 $$1 + 7.92e3T^{2}$$
97 $$1 + 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$