# Properties

 Label 2-1200-60.59-c1-0-8 Degree $2$ Conductor $1200$ Sign $-0.316 - 0.948i$ Analytic cond. $9.58204$ Root an. cond. $3.09548$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.22 + 1.22i)3-s + 2.44·7-s − 2.99i·9-s − 4.89·11-s + 2i·13-s + 6·17-s + 4.89i·19-s + (−2.99 + 2.99i)21-s − 2.44i·23-s + (3.67 + 3.67i)27-s + 9.79i·31-s + (5.99 − 5.99i)33-s + 2i·37-s + (−2.44 − 2.44i)39-s − 6i·41-s + ⋯
 L(s)  = 1 + (−0.707 + 0.707i)3-s + 0.925·7-s − 0.999i·9-s − 1.47·11-s + 0.554i·13-s + 1.45·17-s + 1.12i·19-s + (−0.654 + 0.654i)21-s − 0.510i·23-s + (0.707 + 0.707i)27-s + 1.75i·31-s + (1.04 − 1.04i)33-s + 0.328i·37-s + (−0.392 − 0.392i)39-s − 0.937i·41-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.316 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1200$$    =    $$2^{4} \cdot 3 \cdot 5^{2}$$ Sign: $-0.316 - 0.948i$ Analytic conductor: $$9.58204$$ Root analytic conductor: $$3.09548$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1200} (1199, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1200,\ (\ :1/2),\ -0.316 - 0.948i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.080217738$$ $$L(\frac12)$$ $$\approx$$ $$1.080217738$$ $$L(\frac{3}{2})$$ 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 + (1.22 - 1.22i)T$$
5 $$1$$
good7 $$1 - 2.44T + 7T^{2}$$
11 $$1 + 4.89T + 11T^{2}$$
13 $$1 - 2iT - 13T^{2}$$
17 $$1 - 6T + 17T^{2}$$
19 $$1 - 4.89iT - 19T^{2}$$
23 $$1 + 2.44iT - 23T^{2}$$
29 $$1 - 29T^{2}$$
31 $$1 - 9.79iT - 31T^{2}$$
37 $$1 - 2iT - 37T^{2}$$
41 $$1 + 6iT - 41T^{2}$$
43 $$1 - 2.44T + 43T^{2}$$
47 $$1 - 12.2iT - 47T^{2}$$
53 $$1 + 6T + 53T^{2}$$
59 $$1 + 9.79T + 59T^{2}$$
61 $$1 - 8T + 61T^{2}$$
67 $$1 - 7.34T + 67T^{2}$$
71 $$1 + 4.89T + 71T^{2}$$
73 $$1 - 14iT - 73T^{2}$$
79 $$1 + 4.89iT - 79T^{2}$$
83 $$1 - 7.34iT - 83T^{2}$$
89 $$1 - 12iT - 89T^{2}$$
97 $$1 - 10iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$