# Properties

 Label 2-192-8.5-c3-0-2 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 + 10.3i·5-s − 3.46·7-s − 9·9-s + 55.4i·13-s − 31.1·15-s − 90·17-s − 116i·19-s − 10.3i·21-s − 103.·23-s + 17·25-s − 27i·27-s + 259. i·29-s − 301.·31-s − 36i·35-s + ⋯
 L(s)  = 1 + 0.577i·3-s + 0.929i·5-s − 0.187·7-s − 0.333·9-s + 1.18i·13-s − 0.536·15-s − 1.28·17-s − 1.40i·19-s − 0.107i·21-s − 0.942·23-s + 0.136·25-s − 0.192i·27-s + 1.66i·29-s − 1.74·31-s − 0.173i·35-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

 $$L(2)$$ $$\approx$$ $$0.120766 + 0.917310i$$ $$L(\frac12)$$ $$\approx$$ $$0.120766 + 0.917310i$$ $$L(\frac{5}{2})$$ 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 - 10.3iT - 125T^{2}$$
7 $$1 + 3.46T + 343T^{2}$$
11 $$1 - 1.33e3T^{2}$$
13 $$1 - 55.4iT - 2.19e3T^{2}$$
17 $$1 + 90T + 4.91e3T^{2}$$
19 $$1 + 116iT - 6.85e3T^{2}$$
23 $$1 + 103.T + 1.21e4T^{2}$$
29 $$1 - 259. iT - 2.43e4T^{2}$$
31 $$1 + 301.T + 2.97e4T^{2}$$
37 $$1 - 34.6iT - 5.06e4T^{2}$$
41 $$1 + 54T + 6.89e4T^{2}$$
43 $$1 - 20iT - 7.95e4T^{2}$$
47 $$1 - 394.T + 1.03e5T^{2}$$
53 $$1 - 488. iT - 1.48e5T^{2}$$
59 $$1 - 324iT - 2.05e5T^{2}$$
61 $$1 + 575. iT - 2.26e5T^{2}$$
67 $$1 - 116iT - 3.00e5T^{2}$$
71 $$1 - 1.10e3T + 3.57e5T^{2}$$
73 $$1 - 1.10e3T + 3.89e5T^{2}$$
79 $$1 + 148.T + 4.93e5T^{2}$$
83 $$1 - 1.15e3iT - 5.71e5T^{2}$$
89 $$1 - 918T + 7.04e5T^{2}$$
97 $$1 - 190T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$