# Properties

 Label 2-300-20.19-c2-0-16 Degree $2$ Conductor $300$ Sign $0.786 - 0.617i$ 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 + (1.95 + 0.438i)2-s − 1.73·3-s + (3.61 + 1.71i)4-s + (−3.37 − 0.758i)6-s + 6.33·7-s + (6.30 + 4.92i)8-s + 2.99·9-s + 9.27i·11-s + (−6.26 − 2.96i)12-s − 18.5i·13-s + (12.3 + 2.77i)14-s + (10.1 + 12.3i)16-s + 13.9i·17-s + (5.85 + 1.31i)18-s + 17.2i·19-s + ⋯
 L(s)  = 1 + (0.975 + 0.219i)2-s − 0.577·3-s + (0.904 + 0.427i)4-s + (−0.563 − 0.126i)6-s + 0.904·7-s + (0.788 + 0.615i)8-s + 0.333·9-s + 0.843i·11-s + (−0.521 − 0.246i)12-s − 1.42i·13-s + (0.882 + 0.198i)14-s + (0.634 + 0.772i)16-s + 0.818i·17-s + (0.325 + 0.0730i)18-s + 0.907i·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.786 - 0.617i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.786 - 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$300$$    =    $$2^{2} \cdot 3 \cdot 5^{2}$$ Sign: $0.786 - 0.617i$ Analytic conductor: $$8.17440$$ Root analytic conductor: $$2.85909$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{300} (199, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 300,\ (\ :1),\ 0.786 - 0.617i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$2.63373 + 0.910062i$$ $$L(\frac12)$$ $$\approx$$ $$2.63373 + 0.910062i$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-1.95 - 0.438i)T$$
3 $$1 + 1.73T$$
5 $$1$$
good7 $$1 - 6.33T + 49T^{2}$$
11 $$1 - 9.27iT - 121T^{2}$$
13 $$1 + 18.5iT - 169T^{2}$$
17 $$1 - 13.9iT - 289T^{2}$$
19 $$1 - 17.2iT - 361T^{2}$$
23 $$1 - 33.7T + 529T^{2}$$
29 $$1 - 28.6T + 841T^{2}$$
31 $$1 - 23.4iT - 961T^{2}$$
37 $$1 + 67.3iT - 1.36e3T^{2}$$
41 $$1 + 44.0T + 1.68e3T^{2}$$
43 $$1 + 50.2T + 1.84e3T^{2}$$
47 $$1 + 31.1T + 2.20e3T^{2}$$
53 $$1 + 81.6iT - 2.80e3T^{2}$$
59 $$1 + 19.2iT - 3.48e3T^{2}$$
61 $$1 + 53.1T + 3.72e3T^{2}$$
67 $$1 + 4.49T + 4.48e3T^{2}$$
71 $$1 - 13.3iT - 5.04e3T^{2}$$
73 $$1 + 40.8iT - 5.32e3T^{2}$$
79 $$1 + 141. iT - 6.24e3T^{2}$$
83 $$1 + 69.8T + 6.88e3T^{2}$$
89 $$1 - 46.3T + 7.92e3T^{2}$$
97 $$1 - 68.5iT - 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$