# Properties

 Label 2-384-8.3-c8-0-34 Degree $2$ Conductor $384$ Sign $1$ Analytic cond. $156.433$ Root an. cond. $12.5073$ Motivic weight $8$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 46.7·3-s − 402. i·5-s − 1.93e3i·7-s + 2.18e3·9-s + 2.01e4·11-s + 4.37e4i·13-s − 1.88e4i·15-s + 1.01e5·17-s + 1.39e4·19-s − 9.05e4i·21-s − 4.33e5i·23-s + 2.28e5·25-s + 1.02e5·27-s + 4.67e5i·29-s + 1.48e6i·31-s + ⋯
 L(s)  = 1 + 0.577·3-s − 0.643i·5-s − 0.806i·7-s + 0.333·9-s + 1.37·11-s + 1.53i·13-s − 0.371i·15-s + 1.21·17-s + 0.106·19-s − 0.465i·21-s − 1.54i·23-s + 0.585·25-s + 0.192·27-s + 0.661i·29-s + 1.60i·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(9-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$384$$    =    $$2^{7} \cdot 3$$ Sign: $1$ Analytic conductor: $$156.433$$ Root analytic conductor: $$12.5073$$ Motivic weight: $$8$$ Rational: no Arithmetic: yes Character: $\chi_{384} (319, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 384,\ (\ :4),\ 1)$$

## Particular Values

 $$L(\frac{9}{2})$$ $$\approx$$ $$3.550486315$$ $$L(\frac12)$$ $$\approx$$ $$3.550486315$$ $$L(5)$$ 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 - 46.7T$$
good5 $$1 + 402. iT - 3.90e5T^{2}$$
7 $$1 + 1.93e3iT - 5.76e6T^{2}$$
11 $$1 - 2.01e4T + 2.14e8T^{2}$$
13 $$1 - 4.37e4iT - 8.15e8T^{2}$$
17 $$1 - 1.01e5T + 6.97e9T^{2}$$
19 $$1 - 1.39e4T + 1.69e10T^{2}$$
23 $$1 + 4.33e5iT - 7.83e10T^{2}$$
29 $$1 - 4.67e5iT - 5.00e11T^{2}$$
31 $$1 - 1.48e6iT - 8.52e11T^{2}$$
37 $$1 - 2.57e6iT - 3.51e12T^{2}$$
41 $$1 + 3.84e6T + 7.98e12T^{2}$$
43 $$1 - 1.88e6T + 1.16e13T^{2}$$
47 $$1 - 5.93e6iT - 2.38e13T^{2}$$
53 $$1 + 8.07e6iT - 6.22e13T^{2}$$
59 $$1 + 1.78e7T + 1.46e14T^{2}$$
61 $$1 - 2.11e7iT - 1.91e14T^{2}$$
67 $$1 - 7.11e6T + 4.06e14T^{2}$$
71 $$1 - 1.36e7iT - 6.45e14T^{2}$$
73 $$1 - 3.97e7T + 8.06e14T^{2}$$
79 $$1 - 8.29e5iT - 1.51e15T^{2}$$
83 $$1 + 1.45e7T + 2.25e15T^{2}$$
89 $$1 - 7.77e7T + 3.93e15T^{2}$$
97 $$1 + 9.95e7T + 7.83e15T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$