# Properties

 Degree 2 Conductor $2^{4}$ Sign $-0.824 - 0.565i$ Motivic weight 11 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (30.2 + 33.6i)2-s + (−56.4 + 56.4i)3-s + (−213. + 2.03e3i)4-s + (6.26e3 + 6.26e3i)5-s + (−3.60e3 − 188. i)6-s − 3.29e4i·7-s + (−7.49e4 + 5.45e4i)8-s + 1.70e5i·9-s + (−2.08e4 + 4.00e5i)10-s + (−1.51e5 − 1.51e5i)11-s + (−1.02e5 − 1.27e5i)12-s + (−1.05e6 + 1.05e6i)13-s + (1.10e6 − 9.97e5i)14-s − 7.06e5·15-s + (−4.10e6 − 8.68e5i)16-s − 1.95e6·17-s + ⋯
 L(s)  = 1 + (0.669 + 0.743i)2-s + (−0.134 + 0.134i)3-s + (−0.104 + 0.994i)4-s + (0.895 + 0.895i)5-s + (−0.189 − 0.00989i)6-s − 0.740i·7-s + (−0.808 + 0.588i)8-s + 0.964i·9-s + (−0.0660 + 1.26i)10-s + (−0.284 − 0.284i)11-s + (−0.119 − 0.147i)12-s + (−0.785 + 0.785i)13-s + (0.550 − 0.495i)14-s − 0.240·15-s + (−0.978 − 0.207i)16-s − 0.334·17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.824 - 0.565i)\, \overline{\Lambda}(12-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.824 - 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$16$$    =    $$2^{4}$$ $$\varepsilon$$ = $-0.824 - 0.565i$ motivic weight = $$11$$ character : $\chi_{16} (5, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 16,\ (\ :11/2),\ -0.824 - 0.565i)$ $L(6)$ $\approx$ $0.702127 + 2.26452i$ $L(\frac12)$ $\approx$ $0.702127 + 2.26452i$ $L(\frac{13}{2})$ 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 + (-30.2 - 33.6i)T$$
good3 $$1 + (56.4 - 56.4i)T - 1.77e5iT^{2}$$
5 $$1 + (-6.26e3 - 6.26e3i)T + 4.88e7iT^{2}$$
7 $$1 + 3.29e4iT - 1.97e9T^{2}$$
11 $$1 + (1.51e5 + 1.51e5i)T + 2.85e11iT^{2}$$
13 $$1 + (1.05e6 - 1.05e6i)T - 1.79e12iT^{2}$$
17 $$1 + 1.95e6T + 3.42e13T^{2}$$
19 $$1 + (-5.25e6 + 5.25e6i)T - 1.16e14iT^{2}$$
23 $$1 - 4.62e7iT - 9.52e14T^{2}$$
29 $$1 + (-1.16e8 + 1.16e8i)T - 1.22e16iT^{2}$$
31 $$1 - 2.86e8T + 2.54e16T^{2}$$
37 $$1 + (-3.33e8 - 3.33e8i)T + 1.77e17iT^{2}$$
41 $$1 - 1.31e8iT - 5.50e17T^{2}$$
43 $$1 + (-5.60e8 - 5.60e8i)T + 9.29e17iT^{2}$$
47 $$1 - 1.10e8T + 2.47e18T^{2}$$
53 $$1 + (2.81e9 + 2.81e9i)T + 9.26e18iT^{2}$$
59 $$1 + (3.57e8 + 3.57e8i)T + 3.01e19iT^{2}$$
61 $$1 + (6.15e9 - 6.15e9i)T - 4.35e19iT^{2}$$
67 $$1 + (-8.27e9 + 8.27e9i)T - 1.22e20iT^{2}$$
71 $$1 + 1.51e10iT - 2.31e20T^{2}$$
73 $$1 - 1.11e10iT - 3.13e20T^{2}$$
79 $$1 - 1.68e10T + 7.47e20T^{2}$$
83 $$1 + (1.04e10 - 1.04e10i)T - 1.28e21iT^{2}$$
89 $$1 - 3.00e10iT - 2.77e21T^{2}$$
97 $$1 - 1.53e11T + 7.15e21T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}