# Properties

 Label 2-60e2-20.19-c0-0-5 Degree $2$ Conductor $3600$ Sign $0.834 + 0.550i$ Analytic cond. $1.79663$ Root an. cond. $1.34038$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 1.73·7-s − i·13-s − 1.73i·19-s − 1.73i·31-s + 2i·37-s − 1.73·43-s + 1.99·49-s − 61-s + 1.73·67-s + 2i·73-s − 1.73i·91-s + i·97-s − 109-s + ⋯
 L(s)  = 1 + 1.73·7-s − i·13-s − 1.73i·19-s − 1.73i·31-s + 2i·37-s − 1.73·43-s + 1.99·49-s − 61-s + 1.73·67-s + 2i·73-s − 1.73i·91-s + i·97-s − 109-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.834 + 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.834 + 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3600$$    =    $$2^{4} \cdot 3^{2} \cdot 5^{2}$$ Sign: $0.834 + 0.550i$ Analytic conductor: $$1.79663$$ Root analytic conductor: $$1.34038$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{3600} (1999, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3600,\ (\ :0),\ 0.834 + 0.550i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.549653291$$ $$L(\frac12)$$ $$\approx$$ $$1.549653291$$ $$L(1)$$ 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$$
5 $$1$$
good7 $$1 - 1.73T + T^{2}$$
11 $$1 - T^{2}$$
13 $$1 + iT - T^{2}$$
17 $$1 - T^{2}$$
19 $$1 + 1.73iT - T^{2}$$
23 $$1 + T^{2}$$
29 $$1 + T^{2}$$
31 $$1 + 1.73iT - T^{2}$$
37 $$1 - 2iT - T^{2}$$
41 $$1 + T^{2}$$
43 $$1 + 1.73T + T^{2}$$
47 $$1 + T^{2}$$
53 $$1 - T^{2}$$
59 $$1 - T^{2}$$
61 $$1 + T + T^{2}$$
67 $$1 - 1.73T + T^{2}$$
71 $$1 - T^{2}$$
73 $$1 - 2iT - T^{2}$$
79 $$1 - T^{2}$$
83 $$1 + T^{2}$$
89 $$1 + T^{2}$$
97 $$1 - iT - T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$