# Properties

 Label 2-2300-5.4-c1-0-20 Degree $2$ Conductor $2300$ Sign $0.894 + 0.447i$ Analytic cond. $18.3655$ Root an. cond. $4.28550$ 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 + 3·9-s + 6·11-s − 6i·13-s + 7i·17-s − 2·19-s + i·23-s + 5·29-s + 31-s − 5i·37-s − 7·41-s − 8i·43-s + 8i·47-s + 6·49-s − 3i·53-s + ⋯
 L(s)  = 1 − 0.377i·7-s + 9-s + 1.80·11-s − 1.66i·13-s + 1.69i·17-s − 0.458·19-s + 0.208i·23-s + 0.928·29-s + 0.179·31-s − 0.821i·37-s − 1.09·41-s − 1.21i·43-s + 1.16i·47-s + 0.857·49-s − 0.412i·53-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{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: $$2300$$    =    $$2^{2} \cdot 5^{2} \cdot 23$$ Sign: $0.894 + 0.447i$ Analytic conductor: $$18.3655$$ Root analytic conductor: $$4.28550$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{2300} (1749, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 2300,\ (\ :1/2),\ 0.894 + 0.447i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.192630151$$ $$L(\frac12)$$ $$\approx$$ $$2.192630151$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$1$$
23 $$1 - iT$$
good3 $$1 - 3T^{2}$$
7 $$1 + iT - 7T^{2}$$
11 $$1 - 6T + 11T^{2}$$
13 $$1 + 6iT - 13T^{2}$$
17 $$1 - 7iT - 17T^{2}$$
19 $$1 + 2T + 19T^{2}$$
29 $$1 - 5T + 29T^{2}$$
31 $$1 - T + 31T^{2}$$
37 $$1 + 5iT - 37T^{2}$$
41 $$1 + 7T + 41T^{2}$$
43 $$1 + 8iT - 43T^{2}$$
47 $$1 - 8iT - 47T^{2}$$
53 $$1 + 3iT - 53T^{2}$$
59 $$1 + 13T + 59T^{2}$$
61 $$1 + 8T + 61T^{2}$$
67 $$1 + 9iT - 67T^{2}$$
71 $$1 - 7T + 71T^{2}$$
73 $$1 - 2iT - 73T^{2}$$
79 $$1 - 12T + 79T^{2}$$
83 $$1 - 5iT - 83T^{2}$$
89 $$1 - 12T + 89T^{2}$$
97 $$1 - 2iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$