# Properties

 Degree $2$ Conductor $2100$ Sign $0.894 - 0.447i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + i·3-s + i·7-s − 9-s + 11-s − 4i·13-s + 2i·17-s + 4·19-s − 21-s − 7i·23-s − i·27-s + 9·29-s − 2·31-s + i·33-s + i·37-s + 4·39-s + ⋯
 L(s)  = 1 + 0.577i·3-s + 0.377i·7-s − 0.333·9-s + 0.301·11-s − 1.10i·13-s + 0.485i·17-s + 0.917·19-s − 0.218·21-s − 1.45i·23-s − 0.192i·27-s + 1.67·29-s − 0.359·31-s + 0.174i·33-s + 0.164i·37-s + 0.640·39-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{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: $$2100$$    =    $$2^{2} \cdot 3 \cdot 5^{2} \cdot 7$$ Sign: $0.894 - 0.447i$ Motivic weight: $$1$$ Character: $\chi_{2100} (1849, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 2100,\ (\ :1/2),\ 0.894 - 0.447i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.812408588$$ $$L(\frac12)$$ $$\approx$$ $$1.812408588$$ $$L(\frac{3}{2})$$ 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 - iT$$
5 $$1$$
7 $$1 - iT$$
good11 $$1 - T + 11T^{2}$$
13 $$1 + 4iT - 13T^{2}$$
17 $$1 - 2iT - 17T^{2}$$
19 $$1 - 4T + 19T^{2}$$
23 $$1 + 7iT - 23T^{2}$$
29 $$1 - 9T + 29T^{2}$$
31 $$1 + 2T + 31T^{2}$$
37 $$1 - iT - 37T^{2}$$
41 $$1 - 8T + 41T^{2}$$
43 $$1 - 9iT - 43T^{2}$$
47 $$1 - 4iT - 47T^{2}$$
53 $$1 + 6iT - 53T^{2}$$
59 $$1 + 4T + 59T^{2}$$
61 $$1 - 4T + 61T^{2}$$
67 $$1 - 9iT - 67T^{2}$$
71 $$1 - 5T + 71T^{2}$$
73 $$1 + 10iT - 73T^{2}$$
79 $$1 - 15T + 79T^{2}$$
83 $$1 - 6iT - 83T^{2}$$
89 $$1 + 8T + 89T^{2}$$
97 $$1 - 10iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$