# Properties

 Label 2-1152-8.5-c3-0-39 Degree $2$ Conductor $1152$ Sign $0.707 + 0.707i$ Analytic cond. $67.9702$ Root an. cond. $8.24440$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 8i·5-s + 12·7-s + 12i·11-s − 20i·13-s − 62·17-s − 108i·19-s + 72·23-s + 61·25-s − 128i·29-s − 204·31-s + 96i·35-s − 228i·37-s + 22·41-s − 204i·43-s + 600·47-s + ⋯
 L(s)  = 1 + 0.715i·5-s + 0.647·7-s + 0.328i·11-s − 0.426i·13-s − 0.884·17-s − 1.30i·19-s + 0.652·23-s + 0.487·25-s − 0.819i·29-s − 1.18·31-s + 0.463i·35-s − 1.01i·37-s + 0.0838·41-s − 0.723i·43-s + 1.86·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1152$$    =    $$2^{7} \cdot 3^{2}$$ Sign: $0.707 + 0.707i$ Analytic conductor: $$67.9702$$ Root analytic conductor: $$8.24440$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{1152} (577, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1152,\ (\ :3/2),\ 0.707 + 0.707i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.899880169$$ $$L(\frac12)$$ $$\approx$$ $$1.899880169$$ $$L(\frac{5}{2})$$ 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 - 8iT - 125T^{2}$$
7 $$1 - 12T + 343T^{2}$$
11 $$1 - 12iT - 1.33e3T^{2}$$
13 $$1 + 20iT - 2.19e3T^{2}$$
17 $$1 + 62T + 4.91e3T^{2}$$
19 $$1 + 108iT - 6.85e3T^{2}$$
23 $$1 - 72T + 1.21e4T^{2}$$
29 $$1 + 128iT - 2.43e4T^{2}$$
31 $$1 + 204T + 2.97e4T^{2}$$
37 $$1 + 228iT - 5.06e4T^{2}$$
41 $$1 - 22T + 6.89e4T^{2}$$
43 $$1 + 204iT - 7.95e4T^{2}$$
47 $$1 - 600T + 1.03e5T^{2}$$
53 $$1 + 256iT - 1.48e5T^{2}$$
59 $$1 - 828iT - 2.05e5T^{2}$$
61 $$1 - 84iT - 2.26e5T^{2}$$
67 $$1 + 348iT - 3.00e5T^{2}$$
71 $$1 + 456T + 3.57e5T^{2}$$
73 $$1 - 822T + 3.89e5T^{2}$$
79 $$1 + 1.35e3T + 4.93e5T^{2}$$
83 $$1 - 108iT - 5.71e5T^{2}$$
89 $$1 - 938T + 7.04e5T^{2}$$
97 $$1 - 1.27e3T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$