# Properties

 Label 2-192-4.3-c4-0-12 Degree $2$ Conductor $192$ Sign $i$ Analytic cond. $19.8470$ Root an. cond. $4.45500$ Motivic weight $4$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 5.19i·3-s + 42·5-s − 76.2i·7-s − 27·9-s − 20.7i·11-s + 182·13-s − 218. i·15-s − 246·17-s − 117. i·19-s − 396·21-s + 748. i·23-s + 1.13e3·25-s + 140. i·27-s − 78·29-s − 1.47e3i·31-s + ⋯
 L(s)  = 1 − 0.577i·3-s + 1.67·5-s − 1.55i·7-s − 0.333·9-s − 0.171i·11-s + 1.07·13-s − 0.969i·15-s − 0.851·17-s − 0.326i·19-s − 0.897·21-s + 1.41i·23-s + 1.82·25-s + 0.192i·27-s − 0.0927·29-s − 1.53i·31-s + ⋯

## Functional equation

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

## Invariants

 Degree: $$2$$ Conductor: $$192$$    =    $$2^{6} \cdot 3$$ Sign: $i$ Analytic conductor: $$19.8470$$ Root analytic conductor: $$4.45500$$ Motivic weight: $$4$$ Rational: no Arithmetic: yes Character: $\chi_{192} (127, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 192,\ (\ :2),\ i)$$

## Particular Values

 $$L(\frac{5}{2})$$ $$\approx$$ $$2.456277875$$ $$L(\frac12)$$ $$\approx$$ $$2.456277875$$ $$L(3)$$ 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 + 5.19iT$$
good5 $$1 - 42T + 625T^{2}$$
7 $$1 + 76.2iT - 2.40e3T^{2}$$
11 $$1 + 20.7iT - 1.46e4T^{2}$$
13 $$1 - 182T + 2.85e4T^{2}$$
17 $$1 + 246T + 8.35e4T^{2}$$
19 $$1 + 117. iT - 1.30e5T^{2}$$
23 $$1 - 748. iT - 2.79e5T^{2}$$
29 $$1 + 78T + 7.07e5T^{2}$$
31 $$1 + 1.47e3iT - 9.23e5T^{2}$$
37 $$1 + 530T + 1.87e6T^{2}$$
41 $$1 + 918T + 2.82e6T^{2}$$
43 $$1 + 852. iT - 3.41e6T^{2}$$
47 $$1 + 3.78e3iT - 4.87e6T^{2}$$
53 $$1 - 4.62e3T + 7.89e6T^{2}$$
59 $$1 + 228. iT - 1.21e7T^{2}$$
61 $$1 + 1.34e3T + 1.38e7T^{2}$$
67 $$1 - 1.08e3iT - 2.01e7T^{2}$$
71 $$1 - 1.82e3iT - 2.54e7T^{2}$$
73 $$1 + 926T + 2.83e7T^{2}$$
79 $$1 - 4.39e3iT - 3.89e7T^{2}$$
83 $$1 - 1.19e4iT - 4.74e7T^{2}$$
89 $$1 - 1.15e4T + 6.27e7T^{2}$$
97 $$1 + 1.31e4T + 8.85e7T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$