# Properties

 Label 2-1792-28.27-c1-0-44 Degree $2$ Conductor $1792$ Sign $-0.755 + 0.654i$ Analytic cond. $14.3091$ Root an. cond. $3.78274$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 + 0.732·3-s + 2.73i·5-s + (−2 + 1.73i)7-s − 2.46·9-s − 1.46i·11-s − 1.26i·13-s + 2i·15-s − 4i·17-s − 4.73·19-s + (−1.46 + 1.26i)21-s + 1.46i·23-s − 2.46·25-s − 4·27-s + 6.92·29-s − 6.92·31-s + ⋯
 L(s)  = 1 + 0.422·3-s + 1.22i·5-s + (−0.755 + 0.654i)7-s − 0.821·9-s − 0.441i·11-s − 0.351i·13-s + 0.516i·15-s − 0.970i·17-s − 1.08·19-s + (−0.319 + 0.276i)21-s + 0.305i·23-s − 0.492·25-s − 0.769·27-s + 1.28·29-s − 1.24·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1792$$    =    $$2^{8} \cdot 7$$ Sign: $-0.755 + 0.654i$ Analytic conductor: $$14.3091$$ Root analytic conductor: $$3.78274$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1792} (1791, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$1$$ Selberg data: $$(2,\ 1792,\ (\ :1/2),\ -0.755 + 0.654i)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ 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 - 1.73i)T$$
good3 $$1 - 0.732T + 3T^{2}$$
5 $$1 - 2.73iT - 5T^{2}$$
11 $$1 + 1.46iT - 11T^{2}$$
13 $$1 + 1.26iT - 13T^{2}$$
17 $$1 + 4iT - 17T^{2}$$
19 $$1 + 4.73T + 19T^{2}$$
23 $$1 - 1.46iT - 23T^{2}$$
29 $$1 - 6.92T + 29T^{2}$$
31 $$1 + 6.92T + 31T^{2}$$
37 $$1 + 4T + 37T^{2}$$
41 $$1 - 10.9iT - 41T^{2}$$
43 $$1 + 9.46iT - 43T^{2}$$
47 $$1 - 6.92T + 47T^{2}$$
53 $$1 + 6.92T + 53T^{2}$$
59 $$1 + 14.1T + 59T^{2}$$
61 $$1 + 5.66iT - 61T^{2}$$
67 $$1 + 9.46iT - 67T^{2}$$
71 $$1 + 11.4iT - 71T^{2}$$
73 $$1 + 6.92iT - 73T^{2}$$
79 $$1 - 3.46iT - 79T^{2}$$
83 $$1 + 7.26T + 83T^{2}$$
89 $$1 - 1.07iT - 89T^{2}$$
97 $$1 - 12iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$