Properties

 Label 2-12e3-4.3-c2-0-57 Degree $2$ Conductor $1728$ Sign $-1$ 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 − 6.42·5-s − 13.8i·7-s − 13.0i·11-s − 7.16·13-s + 31.5·17-s − 16.4i·19-s − 16.9i·23-s + 16.2·25-s + 42.4·29-s − 29.6i·31-s + 88.6i·35-s − 39.3·37-s + 39.8·41-s − 16.3i·43-s − 57.8i·47-s + ⋯
 L(s)  = 1 − 1.28·5-s − 1.97i·7-s − 1.18i·11-s − 0.551·13-s + 1.85·17-s − 0.867i·19-s − 0.738i·23-s + 0.649·25-s + 1.46·29-s − 0.957i·31-s + 2.53i·35-s − 1.06·37-s + 0.972·41-s − 0.379i·43-s − 1.23i·47-s + ⋯

Functional equation

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

Invariants

 Degree: $$2$$ Conductor: $$1728$$    =    $$2^{6} \cdot 3^{3}$$ Sign: $-1$ Analytic conductor: $$47.0845$$ Root analytic conductor: $$6.86182$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{1728} (703, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1728,\ (\ :1),\ -1)$$

Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$1.133859906$$ $$L(\frac12)$$ $$\approx$$ $$1.133859906$$ $$L(2)$$ 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 + 6.42T + 25T^{2}$$
7 $$1 + 13.8iT - 49T^{2}$$
11 $$1 + 13.0iT - 121T^{2}$$
13 $$1 + 7.16T + 169T^{2}$$
17 $$1 - 31.5T + 289T^{2}$$
19 $$1 + 16.4iT - 361T^{2}$$
23 $$1 + 16.9iT - 529T^{2}$$
29 $$1 - 42.4T + 841T^{2}$$
31 $$1 + 29.6iT - 961T^{2}$$
37 $$1 + 39.3T + 1.36e3T^{2}$$
41 $$1 - 39.8T + 1.68e3T^{2}$$
43 $$1 + 16.3iT - 1.84e3T^{2}$$
47 $$1 + 57.8iT - 2.20e3T^{2}$$
53 $$1 + 46.4T + 2.80e3T^{2}$$
59 $$1 - 14.2iT - 3.48e3T^{2}$$
61 $$1 - 63.7T + 3.72e3T^{2}$$
67 $$1 - 32.5iT - 4.48e3T^{2}$$
71 $$1 + 22.4iT - 5.04e3T^{2}$$
73 $$1 - 24.9T + 5.32e3T^{2}$$
79 $$1 - 61.9iT - 6.24e3T^{2}$$
83 $$1 - 44.7iT - 6.88e3T^{2}$$
89 $$1 + 1.95T + 7.92e3T^{2}$$
97 $$1 + 44.5T + 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$