Properties

 Label 2-12e3-48.5-c0-0-1 Degree $2$ Conductor $1728$ Sign $0.382 + 0.923i$ Analytic cond. $0.862384$ Root an. cond. $0.928646$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + (0.707 − 0.707i)5-s − i·7-s + (0.707 − 0.707i)11-s + (−1 + i)13-s − 1.41i·17-s − 31-s + (−0.707 − 0.707i)35-s + (1 + i)43-s − 1.41i·47-s + (−0.707 + 0.707i)53-s − 1.00i·55-s + 1.41i·65-s + (−1 + i)67-s + 1.41·71-s − i·73-s + ⋯
 L(s)  = 1 + (0.707 − 0.707i)5-s − i·7-s + (0.707 − 0.707i)11-s + (−1 + i)13-s − 1.41i·17-s − 31-s + (−0.707 − 0.707i)35-s + (1 + i)43-s − 1.41i·47-s + (−0.707 + 0.707i)53-s − 1.00i·55-s + 1.41i·65-s + (−1 + i)67-s + 1.41·71-s − i·73-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$1728$$    =    $$2^{6} \cdot 3^{3}$$ Sign: $0.382 + 0.923i$ Analytic conductor: $$0.862384$$ Root analytic conductor: $$0.928646$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{1728} (1457, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1728,\ (\ :0),\ 0.382 + 0.923i)$$

Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.233905879$$ $$L(\frac12)$$ $$\approx$$ $$1.233905879$$ $$L(1)$$ 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$$
good5 $$1 + (-0.707 + 0.707i)T - iT^{2}$$
7 $$1 + iT - T^{2}$$
11 $$1 + (-0.707 + 0.707i)T - iT^{2}$$
13 $$1 + (1 - i)T - iT^{2}$$
17 $$1 + 1.41iT - T^{2}$$
19 $$1 - iT^{2}$$
23 $$1 + T^{2}$$
29 $$1 + iT^{2}$$
31 $$1 + T + T^{2}$$
37 $$1 + iT^{2}$$
41 $$1 + T^{2}$$
43 $$1 + (-1 - i)T + iT^{2}$$
47 $$1 + 1.41iT - T^{2}$$
53 $$1 + (0.707 - 0.707i)T - iT^{2}$$
59 $$1 - iT^{2}$$
61 $$1 - iT^{2}$$
67 $$1 + (1 - i)T - iT^{2}$$
71 $$1 - 1.41T + T^{2}$$
73 $$1 + iT - T^{2}$$
79 $$1 + T^{2}$$
83 $$1 + (-0.707 - 0.707i)T + iT^{2}$$
89 $$1 - 1.41T + T^{2}$$
97 $$1 - T + T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$