# Properties

 Degree $2$ Conductor $1792$ Sign $0.707 + 0.707i$ Motivic weight $2$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2.55·3-s + 9.86i·5-s + 2.64i·7-s − 2.47·9-s − 13.1·11-s + 5.86i·13-s − 25.2i·15-s − 0.570·17-s − 15.6·19-s − 6.75i·21-s − 16.4i·23-s − 72.3·25-s + 29.3·27-s + 29.7i·29-s − 54.8i·31-s + ⋯
 L(s)  = 1 − 0.851·3-s + 1.97i·5-s + 0.377i·7-s − 0.274·9-s − 1.19·11-s + 0.451i·13-s − 1.68i·15-s − 0.0335·17-s − 0.824·19-s − 0.321i·21-s − 0.716i·23-s − 2.89·25-s + 1.08·27-s + 1.02i·29-s − 1.76i·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1792$$    =    $$2^{8} \cdot 7$$ Sign: $0.707 + 0.707i$ Motivic weight: $$2$$ Character: $\chi_{1792} (127, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1792,\ (\ :1),\ 0.707 + 0.707i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.1591462991$$ $$L(\frac12)$$ $$\approx$$ $$0.1591462991$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
7 $$1 - 2.64iT$$
good3 $$1 + 2.55T + 9T^{2}$$
5 $$1 - 9.86iT - 25T^{2}$$
11 $$1 + 13.1T + 121T^{2}$$
13 $$1 - 5.86iT - 169T^{2}$$
17 $$1 + 0.570T + 289T^{2}$$
19 $$1 + 15.6T + 361T^{2}$$
23 $$1 + 16.4iT - 529T^{2}$$
29 $$1 - 29.7iT - 841T^{2}$$
31 $$1 + 54.8iT - 961T^{2}$$
37 $$1 + 42.0iT - 1.36e3T^{2}$$
41 $$1 + 0.773T + 1.68e3T^{2}$$
43 $$1 + 41.7T + 1.84e3T^{2}$$
47 $$1 - 58.4iT - 2.20e3T^{2}$$
53 $$1 - 5.65iT - 2.80e3T^{2}$$
59 $$1 + 42.6T + 3.48e3T^{2}$$
61 $$1 - 95.9iT - 3.72e3T^{2}$$
67 $$1 - 69.8T + 4.48e3T^{2}$$
71 $$1 - 92.0iT - 5.04e3T^{2}$$
73 $$1 + 9.97T + 5.32e3T^{2}$$
79 $$1 + 20.1iT - 6.24e3T^{2}$$
83 $$1 - 151.T + 6.88e3T^{2}$$
89 $$1 + 5.79T + 7.92e3T^{2}$$
97 $$1 - 103.T + 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$