# Properties

 Label 2-192-12.11-c3-0-2 Degree $2$ Conductor $192$ Sign $-i$ 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 − 5.19i·3-s + 31.1i·7-s − 27·9-s − 70·13-s + 155. i·19-s + 162·21-s + 125·25-s + 140. i·27-s + 155. i·31-s − 110·37-s + 363. i·39-s − 218. i·43-s − 629·49-s + 810·57-s − 182·61-s + ⋯
 L(s)  = 1 − 0.999i·3-s + 1.68i·7-s − 9-s − 1.49·13-s + 1.88i·19-s + 1.68·21-s + 25-s + 1.00i·27-s + 0.903i·31-s − 0.488·37-s + 1.49i·39-s − 0.773i·43-s − 1.83·49-s + 1.88·57-s − 0.382·61-s + ⋯

## Functional equation

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

## Invariants

 Degree: $$2$$ Conductor: $$192$$    =    $$2^{6} \cdot 3$$ Sign: $-i$ Analytic conductor: $$11.3283$$ Root analytic conductor: $$3.36576$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{192} (191, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 192,\ (\ :3/2),\ -i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.649630 + 0.649630i$$ $$L(\frac12)$$ $$\approx$$ $$0.649630 + 0.649630i$$ $$L(\frac{5}{2})$$ 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 + 5.19iT$$
good5 $$1 - 125T^{2}$$
7 $$1 - 31.1iT - 343T^{2}$$
11 $$1 + 1.33e3T^{2}$$
13 $$1 + 70T + 2.19e3T^{2}$$
17 $$1 - 4.91e3T^{2}$$
19 $$1 - 155. iT - 6.85e3T^{2}$$
23 $$1 + 1.21e4T^{2}$$
29 $$1 - 2.43e4T^{2}$$
31 $$1 - 155. iT - 2.97e4T^{2}$$
37 $$1 + 110T + 5.06e4T^{2}$$
41 $$1 - 6.89e4T^{2}$$
43 $$1 + 218. iT - 7.95e4T^{2}$$
47 $$1 + 1.03e5T^{2}$$
53 $$1 - 1.48e5T^{2}$$
59 $$1 + 2.05e5T^{2}$$
61 $$1 + 182T + 2.26e5T^{2}$$
67 $$1 - 654. iT - 3.00e5T^{2}$$
71 $$1 + 3.57e5T^{2}$$
73 $$1 + 1.19e3T + 3.89e5T^{2}$$
79 $$1 + 1.09e3iT - 4.93e5T^{2}$$
83 $$1 + 5.71e5T^{2}$$
89 $$1 - 7.04e5T^{2}$$
97 $$1 - 1.33e3T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$