# Properties

 Label 2-12e3-12.11-c1-0-2 Degree $2$ Conductor $1728$ Sign $-1$ Analytic cond. $13.7981$ Root an. cond. $3.71458$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 2.23i·5-s + 3.87i·7-s − 1.73·11-s − 2·13-s + 4.47i·17-s − 6.92·23-s − 4.47i·29-s − 3.87i·31-s − 8.66·35-s + 4·37-s − 8.94i·41-s + 7.74i·43-s − 3.46·47-s − 8.00·49-s + 2.23i·53-s + ⋯
 L(s)  = 1 + 0.999i·5-s + 1.46i·7-s − 0.522·11-s − 0.554·13-s + 1.08i·17-s − 1.44·23-s − 0.830i·29-s − 0.695i·31-s − 1.46·35-s + 0.657·37-s − 1.39i·41-s + 1.18i·43-s − 0.505·47-s − 1.14·49-s + 0.307i·53-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1728$$    =    $$2^{6} \cdot 3^{3}$$ Sign: $-1$ Analytic conductor: $$13.7981$$ Root analytic conductor: $$3.71458$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1728} (1727, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1728,\ (\ :1/2),\ -1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.8379584553$$ $$L(\frac12)$$ $$\approx$$ $$0.8379584553$$ $$L(\frac{3}{2})$$ 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$$
good5 $$1 - 2.23iT - 5T^{2}$$
7 $$1 - 3.87iT - 7T^{2}$$
11 $$1 + 1.73T + 11T^{2}$$
13 $$1 + 2T + 13T^{2}$$
17 $$1 - 4.47iT - 17T^{2}$$
19 $$1 - 19T^{2}$$
23 $$1 + 6.92T + 23T^{2}$$
29 $$1 + 4.47iT - 29T^{2}$$
31 $$1 + 3.87iT - 31T^{2}$$
37 $$1 - 4T + 37T^{2}$$
41 $$1 + 8.94iT - 41T^{2}$$
43 $$1 - 7.74iT - 43T^{2}$$
47 $$1 + 3.46T + 47T^{2}$$
53 $$1 - 2.23iT - 53T^{2}$$
59 $$1 + 3.46T + 59T^{2}$$
61 $$1 - 4T + 61T^{2}$$
67 $$1 + 7.74iT - 67T^{2}$$
71 $$1 + 10.3T + 71T^{2}$$
73 $$1 - 5T + 73T^{2}$$
79 $$1 + 7.74iT - 79T^{2}$$
83 $$1 + 12.1T + 83T^{2}$$
89 $$1 - 4.47iT - 89T^{2}$$
97 $$1 - 11T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$