# Properties

 Label 2-192-4.3-c8-0-6 Degree $2$ Conductor $192$ Sign $-1$ Analytic cond. $78.2166$ Root an. cond. $8.84402$ Motivic weight $8$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 46.7i·3-s + 722.·5-s + 891. i·7-s − 2.18e3·9-s + 2.57e4i·11-s + 5.89e3·13-s + 3.37e4i·15-s − 1.12e5·17-s − 1.33e5i·19-s − 4.16e4·21-s + 5.38e4i·23-s + 1.31e5·25-s − 1.02e5i·27-s − 1.20e6·29-s + 1.05e6i·31-s + ⋯
 L(s)  = 1 + 0.577i·3-s + 1.15·5-s + 0.371i·7-s − 0.333·9-s + 1.76i·11-s + 0.206·13-s + 0.667i·15-s − 1.34·17-s − 1.02i·19-s − 0.214·21-s + 0.192i·23-s + 0.335·25-s − 0.192i·27-s − 1.71·29-s + 1.14i·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(9-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$192$$    =    $$2^{6} \cdot 3$$ Sign: $-1$ Analytic conductor: $$78.2166$$ Root analytic conductor: $$8.84402$$ Motivic weight: $$8$$ Rational: no Arithmetic: yes Character: $\chi_{192} (127, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 192,\ (\ :4),\ -1)$$

## Particular Values

 $$L(\frac{9}{2})$$ $$\approx$$ $$1.248624124$$ $$L(\frac12)$$ $$\approx$$ $$1.248624124$$ $$L(5)$$ 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 - 46.7iT$$
good5 $$1 - 722.T + 3.90e5T^{2}$$
7 $$1 - 891. iT - 5.76e6T^{2}$$
11 $$1 - 2.57e4iT - 2.14e8T^{2}$$
13 $$1 - 5.89e3T + 8.15e8T^{2}$$
17 $$1 + 1.12e5T + 6.97e9T^{2}$$
19 $$1 + 1.33e5iT - 1.69e10T^{2}$$
23 $$1 - 5.38e4iT - 7.83e10T^{2}$$
29 $$1 + 1.20e6T + 5.00e11T^{2}$$
31 $$1 - 1.05e6iT - 8.52e11T^{2}$$
37 $$1 - 7.11e5T + 3.51e12T^{2}$$
41 $$1 - 4.16e6T + 7.98e12T^{2}$$
43 $$1 - 1.24e6iT - 1.16e13T^{2}$$
47 $$1 - 4.97e4iT - 2.38e13T^{2}$$
53 $$1 + 3.55e6T + 6.22e13T^{2}$$
59 $$1 + 1.36e7iT - 1.46e14T^{2}$$
61 $$1 + 1.96e7T + 1.91e14T^{2}$$
67 $$1 + 5.61e6iT - 4.06e14T^{2}$$
71 $$1 - 2.39e7iT - 6.45e14T^{2}$$
73 $$1 + 3.56e7T + 8.06e14T^{2}$$
79 $$1 - 2.42e6iT - 1.51e15T^{2}$$
83 $$1 + 6.78e7iT - 2.25e15T^{2}$$
89 $$1 + 5.08e7T + 3.93e15T^{2}$$
97 $$1 - 6.06e6T + 7.83e15T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$