# Properties

 Degree 2 Conductor $2^{5} \cdot 3^{2} \cdot 7$ Sign $-0.935 + 0.353i$ Motivic weight 2 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 9.03i·5-s + 2.64i·7-s + 12.4·11-s + 9.03i·13-s − 12.3·17-s − 28.8·19-s + 24.6i·23-s − 56.5·25-s + 22.4i·29-s − 16.7i·31-s − 23.8·35-s − 16.2i·37-s − 6.97·41-s + 22.8·43-s + 6.19i·47-s + ⋯
 L(s)  = 1 + 1.80i·5-s + 0.377i·7-s + 1.13·11-s + 0.694i·13-s − 0.726·17-s − 1.51·19-s + 1.07i·23-s − 2.26·25-s + 0.774i·29-s − 0.541i·31-s − 0.682·35-s − 0.439i·37-s − 0.170·41-s + 0.530·43-s + 0.131i·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 + 0.353i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.935 + 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$2016$$    =    $$2^{5} \cdot 3^{2} \cdot 7$$ $$\varepsilon$$ = $-0.935 + 0.353i$ motivic weight = $$2$$ character : $\chi_{2016} (1135, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 2016,\ (\ :1),\ -0.935 + 0.353i)$$ $$L(\frac{3}{2})$$ $$\approx$$ $$1.095448786$$ $$L(\frac12)$$ $$\approx$$ $$1.095448786$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;7\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
7 $$1 - 2.64iT$$
good5 $$1 - 9.03iT - 25T^{2}$$
11 $$1 - 12.4T + 121T^{2}$$
13 $$1 - 9.03iT - 169T^{2}$$
17 $$1 + 12.3T + 289T^{2}$$
19 $$1 + 28.8T + 361T^{2}$$
23 $$1 - 24.6iT - 529T^{2}$$
29 $$1 - 22.4iT - 841T^{2}$$
31 $$1 + 16.7iT - 961T^{2}$$
37 $$1 + 16.2iT - 1.36e3T^{2}$$
41 $$1 + 6.97T + 1.68e3T^{2}$$
43 $$1 - 22.8T + 1.84e3T^{2}$$
47 $$1 - 6.19iT - 2.20e3T^{2}$$
53 $$1 - 8.01iT - 2.80e3T^{2}$$
59 $$1 - 30.4T + 3.48e3T^{2}$$
61 $$1 - 15.2iT - 3.72e3T^{2}$$
67 $$1 - 78.6T + 4.48e3T^{2}$$
71 $$1 - 17.5iT - 5.04e3T^{2}$$
73 $$1 - 46.6T + 5.32e3T^{2}$$
79 $$1 + 81.0iT - 6.24e3T^{2}$$
83 $$1 - 40.3T + 6.88e3T^{2}$$
89 $$1 + 111.T + 7.92e3T^{2}$$
97 $$1 + 164.T + 9.40e3T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}