Properties

 Label 2-1152-8.3-c0-0-0 Degree $2$ Conductor $1152$ Sign $-i$ Analytic cond. $0.574922$ Root an. cond. $0.758236$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + 2i·5-s − 3·25-s + 2i·29-s + 49-s − 2i·53-s + 2·73-s + 2·97-s − 2i·101-s + ⋯
 L(s)  = 1 + 2i·5-s − 3·25-s + 2i·29-s + 49-s − 2i·53-s + 2·73-s + 2·97-s − 2i·101-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$1152$$    =    $$2^{7} \cdot 3^{2}$$ Sign: $-i$ Analytic conductor: $$0.574922$$ Root analytic conductor: $$0.758236$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{1152} (703, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1152,\ (\ :0),\ -i)$$

Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.9859431593$$ $$L(\frac12)$$ $$\approx$$ $$0.9859431593$$ $$L(1)$$ 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 - 2iT - T^{2}$$
7 $$1 - T^{2}$$
11 $$1 + T^{2}$$
13 $$1 - T^{2}$$
17 $$1 + T^{2}$$
19 $$1 + T^{2}$$
23 $$1 - T^{2}$$
29 $$1 - 2iT - T^{2}$$
31 $$1 - T^{2}$$
37 $$1 - T^{2}$$
41 $$1 + T^{2}$$
43 $$1 + T^{2}$$
47 $$1 - T^{2}$$
53 $$1 + 2iT - T^{2}$$
59 $$1 + T^{2}$$
61 $$1 - T^{2}$$
67 $$1 + T^{2}$$
71 $$1 - T^{2}$$
73 $$1 - 2T + T^{2}$$
79 $$1 - T^{2}$$
83 $$1 + T^{2}$$
89 $$1 + T^{2}$$
97 $$1 - 2T + T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$