# Properties

 Label 2-1200-5.4-c3-0-31 Degree $2$ Conductor $1200$ Sign $0.894 + 0.447i$ Analytic cond. $70.8022$ Root an. cond. $8.41440$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 3i·3-s + 24i·7-s − 9·9-s − 52·11-s − 22i·13-s − 14i·17-s − 20·19-s − 72·21-s − 168i·23-s − 27i·27-s − 230·29-s + 288·31-s − 156i·33-s − 34i·37-s + 66·39-s + ⋯
 L(s)  = 1 + 0.577i·3-s + 1.29i·7-s − 0.333·9-s − 1.42·11-s − 0.469i·13-s − 0.199i·17-s − 0.241·19-s − 0.748·21-s − 1.52i·23-s − 0.192i·27-s − 1.47·29-s + 1.66·31-s − 0.822i·33-s − 0.151i·37-s + 0.270·39-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1200$$    =    $$2^{4} \cdot 3 \cdot 5^{2}$$ Sign: $0.894 + 0.447i$ Analytic conductor: $$70.8022$$ Root analytic conductor: $$8.41440$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{1200} (49, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1200,\ (\ :3/2),\ 0.894 + 0.447i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.171041577$$ $$L(\frac12)$$ $$\approx$$ $$1.171041577$$ $$L(\frac{5}{2})$$ 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 - 24iT - 343T^{2}$$
11 $$1 + 52T + 1.33e3T^{2}$$
13 $$1 + 22iT - 2.19e3T^{2}$$
17 $$1 + 14iT - 4.91e3T^{2}$$
19 $$1 + 20T + 6.85e3T^{2}$$
23 $$1 + 168iT - 1.21e4T^{2}$$
29 $$1 + 230T + 2.43e4T^{2}$$
31 $$1 - 288T + 2.97e4T^{2}$$
37 $$1 + 34iT - 5.06e4T^{2}$$
41 $$1 - 122T + 6.89e4T^{2}$$
43 $$1 + 188iT - 7.95e4T^{2}$$
47 $$1 + 256iT - 1.03e5T^{2}$$
53 $$1 - 338iT - 1.48e5T^{2}$$
59 $$1 - 100T + 2.05e5T^{2}$$
61 $$1 - 742T + 2.26e5T^{2}$$
67 $$1 - 84iT - 3.00e5T^{2}$$
71 $$1 - 328T + 3.57e5T^{2}$$
73 $$1 - 38iT - 3.89e5T^{2}$$
79 $$1 + 240T + 4.93e5T^{2}$$
83 $$1 - 1.21e3iT - 5.71e5T^{2}$$
89 $$1 + 330T + 7.04e5T^{2}$$
97 $$1 - 866iT - 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$