# Properties

 Degree 2 Conductor $2^{3}$ Sign $0.625 + 0.779i$ Motivic weight 20 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−749. + 697. i)2-s − 1.61e4·3-s + (7.62e4 − 1.04e6i)4-s + 8.44e6i·5-s + (1.20e7 − 1.12e7i)6-s + 6.17e7i·7-s + (6.72e8 + 8.37e8i)8-s − 3.22e9·9-s + (−5.89e9 − 6.33e9i)10-s − 2.72e10·11-s + (−1.22e9 + 1.68e10i)12-s + 6.05e10i·13-s + (−4.30e10 − 4.62e10i)14-s − 1.36e11i·15-s + (−1.08e12 − 1.59e11i)16-s − 3.07e11·17-s + ⋯
 L(s)  = 1 + (−0.732 + 0.680i)2-s − 0.272·3-s + (0.0726 − 0.997i)4-s + 0.865i·5-s + (0.199 − 0.185i)6-s + 0.218i·7-s + (0.625 + 0.779i)8-s − 0.925·9-s + (−0.589 − 0.633i)10-s − 1.05·11-s + (−0.0198 + 0.272i)12-s + 0.439i·13-s + (−0.148 − 0.160i)14-s − 0.236i·15-s + (−0.989 − 0.144i)16-s − 0.152·17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.625 + 0.779i)\, \overline{\Lambda}(21-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (0.625 + 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$8$$    =    $$2^{3}$$ $$\varepsilon$$ = $0.625 + 0.779i$ motivic weight = $$20$$ character : $\chi_{8} (3, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 8,\ (\ :10),\ 0.625 + 0.779i)$ $L(\frac{21}{2})$ $\approx$ $0.400597 - 0.192154i$ $L(\frac12)$ $\approx$ $0.400597 - 0.192154i$ $L(11)$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 2$, $$F_p$$ is a polynomial of degree 2. If $p = 2$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 $$1 + (749. - 697. i)T$$
good3 $$1 + 1.61e4T + 3.48e9T^{2}$$
5 $$1 - 8.44e6iT - 9.53e13T^{2}$$
7 $$1 - 6.17e7iT - 7.97e16T^{2}$$
11 $$1 + 2.72e10T + 6.72e20T^{2}$$
13 $$1 - 6.05e10iT - 1.90e22T^{2}$$
17 $$1 + 3.07e11T + 4.06e24T^{2}$$
19 $$1 - 7.50e12T + 3.75e25T^{2}$$
23 $$1 + 5.30e13iT - 1.71e27T^{2}$$
29 $$1 + 5.67e14iT - 1.76e29T^{2}$$
31 $$1 - 1.32e14iT - 6.71e29T^{2}$$
37 $$1 - 2.44e14iT - 2.31e31T^{2}$$
41 $$1 + 1.67e16T + 1.80e32T^{2}$$
43 $$1 + 8.85e14T + 4.67e32T^{2}$$
47 $$1 + 7.52e16iT - 2.76e33T^{2}$$
53 $$1 + 2.98e17iT - 3.05e34T^{2}$$
59 $$1 + 6.61e16T + 2.61e35T^{2}$$
61 $$1 + 1.14e17iT - 5.08e35T^{2}$$
67 $$1 - 1.03e18T + 3.32e36T^{2}$$
71 $$1 - 4.52e18iT - 1.05e37T^{2}$$
73 $$1 + 3.78e18T + 1.84e37T^{2}$$
79 $$1 - 9.29e18iT - 8.96e37T^{2}$$
83 $$1 - 2.33e19T + 2.40e38T^{2}$$
89 $$1 + 4.76e19T + 9.72e38T^{2}$$
97 $$1 + 7.53e19T + 5.43e39T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}