# Properties

 Label 2-300-15.14-c2-0-2 Degree $2$ Conductor $300$ Sign $0.447 - 0.894i$ Analytic cond. $8.17440$ Root an. cond. $2.85909$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 3i·3-s + 13i·7-s − 9·9-s + 23i·13-s − 11·19-s + 39·21-s + 27i·27-s + 59·31-s − 26i·37-s + 69·39-s + 83i·43-s − 120·49-s + 33i·57-s − 121·61-s − 117i·63-s + ⋯
 L(s)  = 1 − i·3-s + 1.85i·7-s − 9-s + 1.76i·13-s − 0.578·19-s + 1.85·21-s + i·27-s + 1.90·31-s − 0.702i·37-s + 1.76·39-s + 1.93i·43-s − 2.44·49-s + 0.578i·57-s − 1.98·61-s − 1.85i·63-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$300$$    =    $$2^{2} \cdot 3 \cdot 5^{2}$$ Sign: $0.447 - 0.894i$ Analytic conductor: $$8.17440$$ Root analytic conductor: $$2.85909$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{300} (149, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 300,\ (\ :1),\ 0.447 - 0.894i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$1.05048 + 0.649234i$$ $$L(\frac12)$$ $$\approx$$ $$1.05048 + 0.649234i$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1 + 3iT$$
5 $$1$$
good7 $$1 - 13iT - 49T^{2}$$
11 $$1 - 121T^{2}$$
13 $$1 - 23iT - 169T^{2}$$
17 $$1 + 289T^{2}$$
19 $$1 + 11T + 361T^{2}$$
23 $$1 + 529T^{2}$$
29 $$1 - 841T^{2}$$
31 $$1 - 59T + 961T^{2}$$
37 $$1 + 26iT - 1.36e3T^{2}$$
41 $$1 - 1.68e3T^{2}$$
43 $$1 - 83iT - 1.84e3T^{2}$$
47 $$1 + 2.20e3T^{2}$$
53 $$1 + 2.80e3T^{2}$$
59 $$1 - 3.48e3T^{2}$$
61 $$1 + 121T + 3.72e3T^{2}$$
67 $$1 - 13iT - 4.48e3T^{2}$$
71 $$1 - 5.04e3T^{2}$$
73 $$1 + 46iT - 5.32e3T^{2}$$
79 $$1 - 142T + 6.24e3T^{2}$$
83 $$1 + 6.88e3T^{2}$$
89 $$1 - 7.92e3T^{2}$$
97 $$1 + 167iT - 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$