# Properties

 Label 2-600-15.8-c1-0-7 Degree $2$ Conductor $600$ Sign $0.920 + 0.391i$ Analytic cond. $4.79102$ Root an. cond. $2.18884$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.70 − 0.292i)3-s + (−2 + 2i)7-s + (2.82 + i)9-s − 5.65i·11-s + (2.82 + 2.82i)17-s + 4i·19-s + (4 − 2.82i)21-s + (4.24 − 4.24i)23-s + (−4.53 − 2.53i)27-s + 5.65·29-s + 8·31-s + (−1.65 + 9.65i)33-s + (8 − 8i)37-s − 5.65i·41-s + (−2 − 2i)43-s + ⋯
 L(s)  = 1 + (−0.985 − 0.169i)3-s + (−0.755 + 0.755i)7-s + (0.942 + 0.333i)9-s − 1.70i·11-s + (0.685 + 0.685i)17-s + 0.917i·19-s + (0.872 − 0.617i)21-s + (0.884 − 0.884i)23-s + (−0.872 − 0.487i)27-s + 1.05·29-s + 1.43·31-s + (−0.288 + 1.68i)33-s + (1.31 − 1.31i)37-s − 0.883i·41-s + (−0.304 − 0.304i)43-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 + 0.391i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.920 + 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$600$$    =    $$2^{3} \cdot 3 \cdot 5^{2}$$ Sign: $0.920 + 0.391i$ Analytic conductor: $$4.79102$$ Root analytic conductor: $$2.18884$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{600} (593, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 600,\ (\ :1/2),\ 0.920 + 0.391i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.961172 - 0.195711i$$ $$L(\frac12)$$ $$\approx$$ $$0.961172 - 0.195711i$$ $$L(\frac{3}{2})$$ 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 + (1.70 + 0.292i)T$$
5 $$1$$
good7 $$1 + (2 - 2i)T - 7iT^{2}$$
11 $$1 + 5.65iT - 11T^{2}$$
13 $$1 + 13iT^{2}$$
17 $$1 + (-2.82 - 2.82i)T + 17iT^{2}$$
19 $$1 - 4iT - 19T^{2}$$
23 $$1 + (-4.24 + 4.24i)T - 23iT^{2}$$
29 $$1 - 5.65T + 29T^{2}$$
31 $$1 - 8T + 31T^{2}$$
37 $$1 + (-8 + 8i)T - 37iT^{2}$$
41 $$1 + 5.65iT - 41T^{2}$$
43 $$1 + (2 + 2i)T + 43iT^{2}$$
47 $$1 + (1.41 + 1.41i)T + 47iT^{2}$$
53 $$1 + (-5.65 + 5.65i)T - 53iT^{2}$$
59 $$1 - 5.65T + 59T^{2}$$
61 $$1 + 6T + 61T^{2}$$
67 $$1 + (6 - 6i)T - 67iT^{2}$$
71 $$1 - 11.3iT - 71T^{2}$$
73 $$1 + (-8 - 8i)T + 73iT^{2}$$
79 $$1 - 79T^{2}$$
83 $$1 + (-9.89 + 9.89i)T - 83iT^{2}$$
89 $$1 + 11.3T + 89T^{2}$$
97 $$1 + (8 - 8i)T - 97iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$