# Properties

 Label 2-192-3.2-c10-0-75 Degree $2$ Conductor $192$ Sign $-0.996 - 0.0775i$ Analytic cond. $121.988$ Root an. cond. $11.0448$ Motivic weight $10$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (242. + 18.8i)3-s − 4.81e3i·5-s − 670.·7-s + (5.83e4 + 9.12e3i)9-s − 2.33e5i·11-s − 3.07e5·13-s + (9.07e4 − 1.16e6i)15-s − 6.72e5i·17-s − 1.55e6·19-s + (−1.62e5 − 1.26e4i)21-s + 5.57e6i·23-s − 1.34e7·25-s + (1.39e7 + 3.30e6i)27-s − 2.97e7i·29-s − 3.09e7·31-s + ⋯
 L(s)  = 1 + (0.996 + 0.0775i)3-s − 1.54i·5-s − 0.0398·7-s + (0.987 + 0.154i)9-s − 1.44i·11-s − 0.828·13-s + (0.119 − 1.53i)15-s − 0.473i·17-s − 0.626·19-s + (−0.0397 − 0.00309i)21-s + 0.866i·23-s − 1.37·25-s + (0.973 + 0.230i)27-s − 1.44i·29-s − 1.08·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0775i)\, \overline{\Lambda}(11-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.996 - 0.0775i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$192$$    =    $$2^{6} \cdot 3$$ Sign: $-0.996 - 0.0775i$ Analytic conductor: $$121.988$$ Root analytic conductor: $$11.0448$$ Motivic weight: $$10$$ Rational: no Arithmetic: yes Character: $\chi_{192} (65, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 192,\ (\ :5),\ -0.996 - 0.0775i)$$

## Particular Values

 $$L(\frac{11}{2})$$ $$\approx$$ $$1.715235644$$ $$L(\frac12)$$ $$\approx$$ $$1.715235644$$ $$L(6)$$ 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 + (-242. - 18.8i)T$$
good5 $$1 + 4.81e3iT - 9.76e6T^{2}$$
7 $$1 + 670.T + 2.82e8T^{2}$$
11 $$1 + 2.33e5iT - 2.59e10T^{2}$$
13 $$1 + 3.07e5T + 1.37e11T^{2}$$
17 $$1 + 6.72e5iT - 2.01e12T^{2}$$
19 $$1 + 1.55e6T + 6.13e12T^{2}$$
23 $$1 - 5.57e6iT - 4.14e13T^{2}$$
29 $$1 + 2.97e7iT - 4.20e14T^{2}$$
31 $$1 + 3.09e7T + 8.19e14T^{2}$$
37 $$1 - 8.56e7T + 4.80e15T^{2}$$
41 $$1 + 3.59e7iT - 1.34e16T^{2}$$
43 $$1 + 3.66e7T + 2.16e16T^{2}$$
47 $$1 - 3.28e7iT - 5.25e16T^{2}$$
53 $$1 - 4.59e8iT - 1.74e17T^{2}$$
59 $$1 - 4.88e8iT - 5.11e17T^{2}$$
61 $$1 - 6.12e7T + 7.13e17T^{2}$$
67 $$1 + 6.70e8T + 1.82e18T^{2}$$
71 $$1 - 1.23e9iT - 3.25e18T^{2}$$
73 $$1 - 1.08e9T + 4.29e18T^{2}$$
79 $$1 - 1.86e9T + 9.46e18T^{2}$$
83 $$1 - 1.09e9iT - 1.55e19T^{2}$$
89 $$1 + 5.19e9iT - 3.11e19T^{2}$$
97 $$1 + 1.07e10T + 7.37e19T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$