# Properties

 Label 2-192-8.5-c3-0-6 Degree $2$ Conductor $192$ Sign $0.965 - 0.258i$ Analytic cond. $11.3283$ Root an. cond. $3.36576$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 3i·3-s − 3.46i·5-s + 24.2·7-s − 9·9-s − 48i·11-s + 41.5i·13-s + 10.3·15-s + 54·17-s − 4i·19-s + 72.7i·21-s + 173.·23-s + 113·25-s − 27i·27-s + 162. i·29-s + 58.8·31-s + ⋯
 L(s)  = 1 + 0.577i·3-s − 0.309i·5-s + 1.30·7-s − 0.333·9-s − 1.31i·11-s + 0.886i·13-s + 0.178·15-s + 0.770·17-s − 0.0482i·19-s + 0.755i·21-s + 1.57·23-s + 0.904·25-s − 0.192i·27-s + 1.04i·29-s + 0.341·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$192$$    =    $$2^{6} \cdot 3$$ Sign: $0.965 - 0.258i$ Analytic conductor: $$11.3283$$ Root analytic conductor: $$3.36576$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{192} (97, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 192,\ (\ :3/2),\ 0.965 - 0.258i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$2.00643 + 0.264152i$$ $$L(\frac12)$$ $$\approx$$ $$2.00643 + 0.264152i$$ $$L(\frac{5}{2})$$ 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 - 3iT$$
good5 $$1 + 3.46iT - 125T^{2}$$
7 $$1 - 24.2T + 343T^{2}$$
11 $$1 + 48iT - 1.33e3T^{2}$$
13 $$1 - 41.5iT - 2.19e3T^{2}$$
17 $$1 - 54T + 4.91e3T^{2}$$
19 $$1 + 4iT - 6.85e3T^{2}$$
23 $$1 - 173.T + 1.21e4T^{2}$$
29 $$1 - 162. iT - 2.43e4T^{2}$$
31 $$1 - 58.8T + 2.97e4T^{2}$$
37 $$1 - 325. iT - 5.06e4T^{2}$$
41 $$1 + 294T + 6.89e4T^{2}$$
43 $$1 + 188iT - 7.95e4T^{2}$$
47 $$1 - 505.T + 1.03e5T^{2}$$
53 $$1 + 744. iT - 1.48e5T^{2}$$
59 $$1 + 252iT - 2.05e5T^{2}$$
61 $$1 + 90.0iT - 2.26e5T^{2}$$
67 $$1 - 628iT - 3.00e5T^{2}$$
71 $$1 + 6.92T + 3.57e5T^{2}$$
73 $$1 + 1.00e3T + 3.89e5T^{2}$$
79 $$1 + 1.34e3T + 4.93e5T^{2}$$
83 $$1 + 720iT - 5.71e5T^{2}$$
89 $$1 + 1.48e3T + 7.04e5T^{2}$$
97 $$1 - 1.82e3T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$