# Properties

 Degree 2 Conductor $2^{3}$ Sign $0.640 - 0.767i$ Motivic weight 13 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (65.8 + 62.0i)2-s − 1.23e3i·3-s + (489. + 8.17e3i)4-s + 2.52e4i·5-s + (7.64e4 − 8.11e4i)6-s + 6.08e5·7-s + (−4.75e5 + 5.69e5i)8-s + 7.79e4·9-s + (−1.56e6 + 1.66e6i)10-s + 7.12e6i·11-s + (1.00e7 − 6.02e5i)12-s − 1.12e7i·13-s + (4.00e7 + 3.77e7i)14-s + 3.11e7·15-s + (−6.66e7 + 8.00e6i)16-s − 5.09e7·17-s + ⋯
 L(s)  = 1 + (0.727 + 0.685i)2-s − 0.975i·3-s + (0.0597 + 0.998i)4-s + 0.723i·5-s + (0.668 − 0.709i)6-s + 1.95·7-s + (−0.640 + 0.767i)8-s + 0.0488·9-s + (−0.495 + 0.526i)10-s + 1.21i·11-s + (0.973 − 0.0582i)12-s − 0.648i·13-s + (1.42 + 1.33i)14-s + 0.705·15-s + (−0.992 + 0.119i)16-s − 0.511·17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.640 - 0.767i)\, \overline{\Lambda}(14-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.640 - 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$8$$    =    $$2^{3}$$ $$\varepsilon$$ = $0.640 - 0.767i$ motivic weight = $$13$$ character : $\chi_{8} (5, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 8,\ (\ :13/2),\ 0.640 - 0.767i)$ $L(7)$ $\approx$ $2.55512 + 1.19527i$ $L(\frac12)$ $\approx$ $2.55512 + 1.19527i$ $L(\frac{15}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 2$, $$F_p(T)$$ is a polynomial of degree 2. If $p = 2$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 + (-65.8 - 62.0i)T$$
good3 $$1 + 1.23e3iT - 1.59e6T^{2}$$
5 $$1 - 2.52e4iT - 1.22e9T^{2}$$
7 $$1 - 6.08e5T + 9.68e10T^{2}$$
11 $$1 - 7.12e6iT - 3.45e13T^{2}$$
13 $$1 + 1.12e7iT - 3.02e14T^{2}$$
17 $$1 + 5.09e7T + 9.90e15T^{2}$$
19 $$1 + 1.86e8iT - 4.20e16T^{2}$$
23 $$1 + 3.07e8T + 5.04e17T^{2}$$
29 $$1 - 4.74e8iT - 1.02e19T^{2}$$
31 $$1 + 2.41e9T + 2.44e19T^{2}$$
37 $$1 + 1.55e10iT - 2.43e20T^{2}$$
41 $$1 + 2.32e10T + 9.25e20T^{2}$$
43 $$1 - 6.55e10iT - 1.71e21T^{2}$$
47 $$1 + 6.84e10T + 5.46e21T^{2}$$
53 $$1 + 9.64e10iT - 2.60e22T^{2}$$
59 $$1 + 3.22e11iT - 1.04e23T^{2}$$
61 $$1 + 6.66e11iT - 1.61e23T^{2}$$
67 $$1 - 5.91e11iT - 5.48e23T^{2}$$
71 $$1 - 8.43e11T + 1.16e24T^{2}$$
73 $$1 + 7.21e11T + 1.67e24T^{2}$$
79 $$1 - 5.78e11T + 4.66e24T^{2}$$
83 $$1 + 1.54e11iT - 8.87e24T^{2}$$
89 $$1 - 2.38e12T + 2.19e25T^{2}$$
97 $$1 + 1.26e13T + 6.73e25T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}