# Properties

 Degree 2 Conductor $2^{3}$ Sign $0.367 + 0.930i$ Motivic weight 13 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−83.4 − 35.1i)2-s − 622. i·3-s + (5.72e3 + 5.85e3i)4-s + 2.66e4i·5-s + (−2.18e4 + 5.19e4i)6-s + 5.04e4·7-s + (−2.72e5 − 6.89e5i)8-s + 1.20e6·9-s + (9.36e5 − 2.22e6i)10-s − 9.06e6i·11-s + (3.64e6 − 3.56e6i)12-s − 8.06e6i·13-s + (−4.21e6 − 1.77e6i)14-s + 1.65e7·15-s + (−1.49e6 + 6.70e7i)16-s + 5.68e7·17-s + ⋯
 L(s)  = 1 + (−0.921 − 0.387i)2-s − 0.492i·3-s + (0.699 + 0.714i)4-s + 0.763i·5-s + (−0.191 + 0.454i)6-s + 0.162·7-s + (−0.367 − 0.930i)8-s + 0.757·9-s + (0.296 − 0.703i)10-s − 1.54i·11-s + (0.352 − 0.344i)12-s − 0.463i·13-s + (−0.149 − 0.0628i)14-s + 0.376·15-s + (−0.0222 + 0.999i)16-s + 0.570·17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.367 + 0.930i)\, \overline{\Lambda}(14-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.367 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$8$$    =    $$2^{3}$$ $$\varepsilon$$ = $0.367 + 0.930i$ motivic weight = $$13$$ character : $\chi_{8} (5, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 8,\ (\ :13/2),\ 0.367 + 0.930i)$ $L(7)$ $\approx$ $0.958609 - 0.652172i$ $L(\frac12)$ $\approx$ $0.958609 - 0.652172i$ $L(\frac{15}{2})$ 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 + (83.4 + 35.1i)T$$
good3 $$1 + 622. iT - 1.59e6T^{2}$$
5 $$1 - 2.66e4iT - 1.22e9T^{2}$$
7 $$1 - 5.04e4T + 9.68e10T^{2}$$
11 $$1 + 9.06e6iT - 3.45e13T^{2}$$
13 $$1 + 8.06e6iT - 3.02e14T^{2}$$
17 $$1 - 5.68e7T + 9.90e15T^{2}$$
19 $$1 + 1.06e8iT - 4.20e16T^{2}$$
23 $$1 - 6.31e8T + 5.04e17T^{2}$$
29 $$1 + 4.06e9iT - 1.02e19T^{2}$$
31 $$1 - 5.86e9T + 2.44e19T^{2}$$
37 $$1 - 2.97e10iT - 2.43e20T^{2}$$
41 $$1 + 5.32e10T + 9.25e20T^{2}$$
43 $$1 + 8.32e9iT - 1.71e21T^{2}$$
47 $$1 - 1.06e11T + 5.46e21T^{2}$$
53 $$1 + 1.04e11iT - 2.60e22T^{2}$$
59 $$1 + 3.49e11iT - 1.04e23T^{2}$$
61 $$1 - 2.73e11iT - 1.61e23T^{2}$$
67 $$1 - 4.69e11iT - 5.48e23T^{2}$$
71 $$1 + 4.62e11T + 1.16e24T^{2}$$
73 $$1 - 4.02e11T + 1.67e24T^{2}$$
79 $$1 + 3.34e12T + 4.66e24T^{2}$$
83 $$1 - 4.78e12iT - 8.87e24T^{2}$$
89 $$1 - 3.03e12T + 2.19e25T^{2}$$
97 $$1 - 4.70e12T + 6.73e25T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}