# Properties

 Label 2-12e3-8.3-c2-0-7 Degree $2$ Conductor $1728$ Sign $-0.707 + 0.707i$ Analytic cond. $47.0845$ Root an. cond. $6.86182$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 6.63i·5-s + 13.4i·7-s − 18.4·11-s − 18.9·25-s + 29.3i·29-s + 61.4i·31-s − 89.4·35-s − 132.·49-s − 105. i·53-s − 122. i·55-s + 117.·59-s + 143.·73-s − 248. i·77-s + 58i·79-s − 165.·83-s + ⋯
 L(s)  = 1 + 1.32i·5-s + 1.92i·7-s − 1.67·11-s − 0.758·25-s + 1.01i·29-s + 1.98i·31-s − 2.55·35-s − 2.71·49-s − 1.99i·53-s − 2.22i·55-s + 1.99·59-s + 1.96·73-s − 3.23i·77-s + 0.734i·79-s − 1.98·83-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1728$$    =    $$2^{6} \cdot 3^{3}$$ Sign: $-0.707 + 0.707i$ Analytic conductor: $$47.0845$$ Root analytic conductor: $$6.86182$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{1728} (1567, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1728,\ (\ :1),\ -0.707 + 0.707i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.9200338645$$ $$L(\frac12)$$ $$\approx$$ $$0.9200338645$$ $$L(2)$$ 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 - 6.63iT - 25T^{2}$$
7 $$1 - 13.4iT - 49T^{2}$$
11 $$1 + 18.4T + 121T^{2}$$
13 $$1 - 169T^{2}$$
17 $$1 + 289T^{2}$$
19 $$1 + 361T^{2}$$
23 $$1 - 529T^{2}$$
29 $$1 - 29.3iT - 841T^{2}$$
31 $$1 - 61.4iT - 961T^{2}$$
37 $$1 - 1.36e3T^{2}$$
41 $$1 + 1.68e3T^{2}$$
43 $$1 + 1.84e3T^{2}$$
47 $$1 - 2.20e3T^{2}$$
53 $$1 + 105. iT - 2.80e3T^{2}$$
59 $$1 - 117.T + 3.48e3T^{2}$$
61 $$1 - 3.72e3T^{2}$$
67 $$1 + 4.48e3T^{2}$$
71 $$1 - 5.04e3T^{2}$$
73 $$1 - 143.T + 5.32e3T^{2}$$
79 $$1 - 58iT - 6.24e3T^{2}$$
83 $$1 + 165.T + 6.88e3T^{2}$$
89 $$1 + 7.92e3T^{2}$$
97 $$1 - 128.T + 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$