# Properties

 Degree $2$ Conductor $16$ Sign $0.0846 + 0.996i$ Motivic weight $4$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (2.47 − 3.14i)2-s + (−7.86 − 7.86i)3-s + (−3.78 − 15.5i)4-s + (27.2 + 27.2i)5-s + (−44.1 + 5.30i)6-s + 50.3·7-s + (−58.2 − 26.5i)8-s + 42.8i·9-s + (152. − 18.3i)10-s + (−53.1 + 53.1i)11-s + (−92.5 + 152. i)12-s + (−125. + 125. i)13-s + (124. − 158. i)14-s − 428. i·15-s + (−227. + 117. i)16-s + 286.·17-s + ⋯
 L(s)  = 1 + (0.617 − 0.786i)2-s + (−0.874 − 0.874i)3-s + (−0.236 − 0.971i)4-s + (1.08 + 1.08i)5-s + (−1.22 + 0.147i)6-s + 1.02·7-s + (−0.910 − 0.414i)8-s + 0.528i·9-s + (1.52 − 0.183i)10-s + (−0.438 + 0.438i)11-s + (−0.642 + 1.05i)12-s + (−0.740 + 0.740i)13-s + (0.634 − 0.807i)14-s − 1.90i·15-s + (−0.888 + 0.459i)16-s + 0.990·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0846 + 0.996i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.0846 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$16$$    =    $$2^{4}$$ Sign: $0.0846 + 0.996i$ Motivic weight: $$4$$ Character: $\chi_{16} (11, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 16,\ (\ :2),\ 0.0846 + 0.996i)$$

## Particular Values

 $$L(\frac{5}{2})$$ $$\approx$$ $$1.02804 - 0.944381i$$ $$L(\frac12)$$ $$\approx$$ $$1.02804 - 0.944381i$$ $$L(3)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-2.47 + 3.14i)T$$
good3 $$1 + (7.86 + 7.86i)T + 81iT^{2}$$
5 $$1 + (-27.2 - 27.2i)T + 625iT^{2}$$
7 $$1 - 50.3T + 2.40e3T^{2}$$
11 $$1 + (53.1 - 53.1i)T - 1.46e4iT^{2}$$
13 $$1 + (125. - 125. i)T - 2.85e4iT^{2}$$
17 $$1 - 286.T + 8.35e4T^{2}$$
19 $$1 + (99.5 + 99.5i)T + 1.30e5iT^{2}$$
23 $$1 + 100.T + 2.79e5T^{2}$$
29 $$1 + (-343. + 343. i)T - 7.07e5iT^{2}$$
31 $$1 - 208. iT - 9.23e5T^{2}$$
37 $$1 + (1.15e3 + 1.15e3i)T + 1.87e6iT^{2}$$
41 $$1 + 2.33e3iT - 2.82e6T^{2}$$
43 $$1 + (2.07e3 - 2.07e3i)T - 3.41e6iT^{2}$$
47 $$1 + 1.05e3iT - 4.87e6T^{2}$$
53 $$1 + (-2.13e3 - 2.13e3i)T + 7.89e6iT^{2}$$
59 $$1 + (-3.72e3 + 3.72e3i)T - 1.21e7iT^{2}$$
61 $$1 + (-2.49e3 + 2.49e3i)T - 1.38e7iT^{2}$$
67 $$1 + (329. + 329. i)T + 2.01e7iT^{2}$$
71 $$1 + 1.04e3T + 2.54e7T^{2}$$
73 $$1 - 2.67e3iT - 2.83e7T^{2}$$
79 $$1 - 4.47e3iT - 3.89e7T^{2}$$
83 $$1 + (-1.45e3 - 1.45e3i)T + 4.74e7iT^{2}$$
89 $$1 + 1.14e3iT - 6.27e7T^{2}$$
97 $$1 + 1.31e4T + 8.85e7T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$