# Properties

 Label 2-30e2-5.4-c1-0-0 Degree $2$ Conductor $900$ Sign $-0.894 - 0.447i$ Analytic cond. $7.18653$ Root an. cond. $2.68077$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + i·7-s − 6·11-s + 5i·13-s − 6i·17-s − 5·19-s + 6i·23-s − 6·29-s − 31-s − 2i·37-s − i·43-s + 6i·47-s + 6·49-s + 12i·53-s − 6·59-s − 13·61-s + ⋯
 L(s)  = 1 + 0.377i·7-s − 1.80·11-s + 1.38i·13-s − 1.45i·17-s − 1.14·19-s + 1.25i·23-s − 1.11·29-s − 0.179·31-s − 0.328i·37-s − 0.152i·43-s + 0.875i·47-s + 0.857·49-s + 1.64i·53-s − 0.781·59-s − 1.66·61-s + ⋯

## Functional equation

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

## Invariants

 Degree: $$2$$ Conductor: $$900$$    =    $$2^{2} \cdot 3^{2} \cdot 5^{2}$$ Sign: $-0.894 - 0.447i$ Analytic conductor: $$7.18653$$ Root analytic conductor: $$2.68077$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{900} (649, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 900,\ (\ :1/2),\ -0.894 - 0.447i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.101794 + 0.431210i$$ $$L(\frac12)$$ $$\approx$$ $$0.101794 + 0.431210i$$ $$L(\frac{3}{2})$$ 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 - iT - 7T^{2}$$
11 $$1 + 6T + 11T^{2}$$
13 $$1 - 5iT - 13T^{2}$$
17 $$1 + 6iT - 17T^{2}$$
19 $$1 + 5T + 19T^{2}$$
23 $$1 - 6iT - 23T^{2}$$
29 $$1 + 6T + 29T^{2}$$
31 $$1 + T + 31T^{2}$$
37 $$1 + 2iT - 37T^{2}$$
41 $$1 + 41T^{2}$$
43 $$1 + iT - 43T^{2}$$
47 $$1 - 6iT - 47T^{2}$$
53 $$1 - 12iT - 53T^{2}$$
59 $$1 + 6T + 59T^{2}$$
61 $$1 + 13T + 61T^{2}$$
67 $$1 + 11iT - 67T^{2}$$
71 $$1 + 71T^{2}$$
73 $$1 - 2iT - 73T^{2}$$
79 $$1 + 8T + 79T^{2}$$
83 $$1 - 6iT - 83T^{2}$$
89 $$1 + 89T^{2}$$
97 $$1 - 7iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$