# Properties

 Degree 2 Conductor $2^{4}$ Sign $0.499 + 0.866i$ Motivic weight 3 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.460 − 2.79i)2-s + (0.756 − 0.756i)3-s + (−7.57 − 2.57i)4-s + (8.22 + 8.22i)5-s + (−1.76 − 2.46i)6-s + 2.67i·7-s + (−10.6 + 19.9i)8-s + 25.8i·9-s + (26.7 − 19.1i)10-s + (−45.2 − 45.2i)11-s + (−7.67 + 3.78i)12-s + (35.3 − 35.3i)13-s + (7.45 + 1.23i)14-s + 12.4·15-s + (50.7 + 38.9i)16-s − 72.4·17-s + ⋯
 L(s)  = 1 + (0.162 − 0.986i)2-s + (0.145 − 0.145i)3-s + (−0.946 − 0.321i)4-s + (0.735 + 0.735i)5-s + (−0.119 − 0.167i)6-s + 0.144i·7-s + (−0.471 + 0.881i)8-s + 0.957i·9-s + (0.845 − 0.605i)10-s + (−1.23 − 1.23i)11-s + (−0.184 + 0.0910i)12-s + (0.755 − 0.755i)13-s + (0.142 + 0.0235i)14-s + 0.214·15-s + (0.793 + 0.609i)16-s − 1.03·17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.499 + 0.866i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.499 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$16$$    =    $$2^{4}$$ $$\varepsilon$$ = $0.499 + 0.866i$ motivic weight = $$3$$ character : $\chi_{16} (5, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 16,\ (\ :3/2),\ 0.499 + 0.866i)$ $L(2)$ $\approx$ $0.943664 - 0.545046i$ $L(\frac12)$ $\approx$ $0.943664 - 0.545046i$ $L(\frac{5}{2})$ 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 + (-0.460 + 2.79i)T$$
good3 $$1 + (-0.756 + 0.756i)T - 27iT^{2}$$
5 $$1 + (-8.22 - 8.22i)T + 125iT^{2}$$
7 $$1 - 2.67iT - 343T^{2}$$
11 $$1 + (45.2 + 45.2i)T + 1.33e3iT^{2}$$
13 $$1 + (-35.3 + 35.3i)T - 2.19e3iT^{2}$$
17 $$1 + 72.4T + 4.91e3T^{2}$$
19 $$1 + (-19.4 + 19.4i)T - 6.85e3iT^{2}$$
23 $$1 - 139. iT - 1.21e4T^{2}$$
29 $$1 + (-66.0 + 66.0i)T - 2.43e4iT^{2}$$
31 $$1 - 188.T + 2.97e4T^{2}$$
37 $$1 + (84.0 + 84.0i)T + 5.06e4iT^{2}$$
41 $$1 + 104. iT - 6.89e4T^{2}$$
43 $$1 + (31.4 + 31.4i)T + 7.95e4iT^{2}$$
47 $$1 + 488.T + 1.03e5T^{2}$$
53 $$1 + (-149. - 149. i)T + 1.48e5iT^{2}$$
59 $$1 + (-284. - 284. i)T + 2.05e5iT^{2}$$
61 $$1 + (228. - 228. i)T - 2.26e5iT^{2}$$
67 $$1 + (-139. + 139. i)T - 3.00e5iT^{2}$$
71 $$1 - 453. iT - 3.57e5T^{2}$$
73 $$1 - 259. iT - 3.89e5T^{2}$$
79 $$1 - 323.T + 4.93e5T^{2}$$
83 $$1 + (563. - 563. i)T - 5.71e5iT^{2}$$
89 $$1 + 866. iT - 7.04e5T^{2}$$
97 $$1 + 936.T + 9.12e5T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}