# Properties

 Degree 2 Conductor $2^{4}$ Sign $0.507 - 0.861i$ Motivic weight 11 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−42.3 + 15.8i)2-s + (−435. + 435. i)3-s + (1.54e3 − 1.34e3i)4-s + (−1.51e3 − 1.51e3i)5-s + (1.15e4 − 2.53e4i)6-s − 5.97e4i·7-s + (−4.41e4 + 8.14e4i)8-s − 2.01e5i·9-s + (8.83e4 + 4.02e4i)10-s + (5.61e4 + 5.61e4i)11-s + (−8.76e4 + 1.25e6i)12-s + (−3.15e5 + 3.15e5i)13-s + (9.46e5 + 2.53e6i)14-s + 1.32e6·15-s + (5.81e5 − 4.15e6i)16-s + 8.11e6·17-s + ⋯
 L(s)  = 1 + (−0.936 + 0.350i)2-s + (−1.03 + 1.03i)3-s + (0.754 − 0.656i)4-s + (−0.217 − 0.217i)5-s + (0.606 − 1.33i)6-s − 1.34i·7-s + (−0.476 + 0.878i)8-s − 1.13i·9-s + (0.279 + 0.127i)10-s + (0.105 + 0.105i)11-s + (−0.101 + 1.45i)12-s + (−0.235 + 0.235i)13-s + (0.470 + 1.25i)14-s + 0.448·15-s + (0.138 − 0.990i)16-s + 1.38·17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.507 - 0.861i)\, \overline{\Lambda}(12-s) \end{aligned}\n
\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.507 - 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\n

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$16$$    =    $$2^{4}$$ $$\varepsilon$$ = $0.507 - 0.861i$ motivic weight = $$11$$ character : $\chi_{16} (5, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 16,\ (\ :11/2),\ 0.507 - 0.861i)$ $L(6)$ $\approx$ $0.576376 + 0.329594i$ $L(\frac12)$ $\approx$ $0.576376 + 0.329594i$ $L(\frac{13}{2})$ 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 + (42.3 - 15.8i)T$$
good3 $$1 + (435. - 435. i)T - 1.77e5iT^{2}$$
5 $$1 + (1.51e3 + 1.51e3i)T + 4.88e7iT^{2}$$
7 $$1 + 5.97e4iT - 1.97e9T^{2}$$
11 $$1 + (-5.61e4 - 5.61e4i)T + 2.85e11iT^{2}$$
13 $$1 + (3.15e5 - 3.15e5i)T - 1.79e12iT^{2}$$
17 $$1 - 8.11e6T + 3.42e13T^{2}$$
19 $$1 + (1.33e7 - 1.33e7i)T - 1.16e14iT^{2}$$
23 $$1 - 2.84e7iT - 9.52e14T^{2}$$
29 $$1 + (-7.59e7 + 7.59e7i)T - 1.22e16iT^{2}$$
31 $$1 - 5.67e7T + 2.54e16T^{2}$$
37 $$1 + (-3.75e8 - 3.75e8i)T + 1.77e17iT^{2}$$
41 $$1 + 1.51e7iT - 5.50e17T^{2}$$
43 $$1 + (-1.08e9 - 1.08e9i)T + 9.29e17iT^{2}$$
47 $$1 - 2.79e9T + 2.47e18T^{2}$$
53 $$1 + (-2.03e9 - 2.03e9i)T + 9.26e18iT^{2}$$
59 $$1 + (4.23e9 + 4.23e9i)T + 3.01e19iT^{2}$$
61 $$1 + (-6.54e9 + 6.54e9i)T - 4.35e19iT^{2}$$
67 $$1 + (8.01e9 - 8.01e9i)T - 1.22e20iT^{2}$$
71 $$1 - 1.40e10iT - 2.31e20T^{2}$$
73 $$1 + 6.29e9iT - 3.13e20T^{2}$$
79 $$1 + 3.46e10T + 7.47e20T^{2}$$
83 $$1 + (-4.29e10 + 4.29e10i)T - 1.28e21iT^{2}$$
89 $$1 + 3.39e9iT - 2.77e21T^{2}$$
97 $$1 - 1.55e10T + 7.15e21T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}