Properties

 Degree 2 Conductor $2^{5} \cdot 3^{2} \cdot 7$ Sign $-0.707 - 0.707i$ Motivic weight 2 Primitive yes Self-dual no Analytic rank 0

Related objects

Dirichlet series

 L(s)  = 1 − 9.86·5-s − 2.64i·7-s − 13.1i·11-s − 5.86·13-s + 0.570·17-s − 15.6i·19-s − 16.4i·23-s + 72.3·25-s + 29.7·29-s − 54.8i·31-s + 26.1i·35-s − 42.0·37-s − 0.773·41-s + 41.7i·43-s − 58.4i·47-s + ⋯
 L(s)  = 1 − 1.97·5-s − 0.377i·7-s − 1.19i·11-s − 0.451·13-s + 0.0335·17-s − 0.824i·19-s − 0.716i·23-s + 2.89·25-s + 1.02·29-s − 1.76i·31-s + 0.745i·35-s − 1.13·37-s − 0.0188·41-s + 0.971i·43-s − 1.24i·47-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 $$d$$ = $$2$$ $$N$$ = $$2016$$    =    $$2^{5} \cdot 3^{2} \cdot 7$$ $$\varepsilon$$ = $-0.707 - 0.707i$ motivic weight = $$2$$ character : $\chi_{2016} (127, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 2016,\ (\ :1),\ -0.707 - 0.707i)$$ $$L(\frac{3}{2})$$ $$\approx$$ $$0.2290112121$$ $$L(\frac12)$$ $$\approx$$ $$0.2290112121$$ $$L(2)$$ 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 + 9.86T + 25T^{2}$$
11 $$1 + 13.1iT - 121T^{2}$$
13 $$1 + 5.86T + 169T^{2}$$
17 $$1 - 0.570T + 289T^{2}$$
19 $$1 + 15.6iT - 361T^{2}$$
23 $$1 + 16.4iT - 529T^{2}$$
29 $$1 - 29.7T + 841T^{2}$$
31 $$1 + 54.8iT - 961T^{2}$$
37 $$1 + 42.0T + 1.36e3T^{2}$$
41 $$1 + 0.773T + 1.68e3T^{2}$$
43 $$1 - 41.7iT - 1.84e3T^{2}$$
47 $$1 + 58.4iT - 2.20e3T^{2}$$
53 $$1 + 5.65T + 2.80e3T^{2}$$
59 $$1 + 42.6iT - 3.48e3T^{2}$$
61 $$1 + 95.9T + 3.72e3T^{2}$$
67 $$1 - 69.8iT - 4.48e3T^{2}$$
71 $$1 - 92.0iT - 5.04e3T^{2}$$
73 $$1 - 9.97T + 5.32e3T^{2}$$
79 $$1 + 20.1iT - 6.24e3T^{2}$$
83 $$1 + 151. iT - 6.88e3T^{2}$$
89 $$1 + 5.79T + 7.92e3T^{2}$$
97 $$1 - 103.T + 9.40e3T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}