# Properties

 Degree 2 Conductor $2 \cdot 3^{2} \cdot 7 \cdot 11$ Sign $-0.820 + 0.572i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (1.37 − 2.37i)5-s + (−1.37 − 2.26i)7-s − 0.999·8-s + (−1.37 − 2.37i)10-s + (0.5 + 0.866i)11-s + 5.49·13-s + (−2.64 + 0.0585i)14-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (4.01 − 6.96i)19-s − 2.74·20-s + 0.999·22-s + (−0.645 + 1.11i)23-s + ⋯
 L(s)  = 1 + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.614 − 1.06i)5-s + (−0.519 − 0.854i)7-s − 0.353·8-s + (−0.434 − 0.752i)10-s + (0.150 + 0.261i)11-s + 1.52·13-s + (−0.706 + 0.0156i)14-s + (−0.125 + 0.216i)16-s + (0.121 + 0.210i)17-s + (0.921 − 1.59i)19-s − 0.614·20-s + 0.213·22-s + (−0.134 + 0.232i)23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.820 + 0.572i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.820 + 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$1386$$    =    $$2 \cdot 3^{2} \cdot 7 \cdot 11$$ $$\varepsilon$$ = $-0.820 + 0.572i$ motivic weight = $$1$$ character : $\chi_{1386} (793, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 1386,\ (\ :1/2),\ -0.820 + 0.572i)$$ $$L(1)$$ $$\approx$$ $$2.058532104$$ $$L(\frac12)$$ $$\approx$$ $$2.058532104$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;7,\;11\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 + (-0.5 + 0.866i)T$$
3 $$1$$
7 $$1 + (1.37 + 2.26i)T$$
11 $$1 + (-0.5 - 0.866i)T$$
good5 $$1 + (-1.37 + 2.37i)T + (-2.5 - 4.33i)T^{2}$$
13 $$1 - 5.49T + 13T^{2}$$
17 $$1 + (-0.5 - 0.866i)T + (-8.5 + 14.7i)T^{2}$$
19 $$1 + (-4.01 + 6.96i)T + (-9.5 - 16.4i)T^{2}$$
23 $$1 + (0.645 - 1.11i)T + (-11.5 - 19.9i)T^{2}$$
29 $$1 + 4.54T + 29T^{2}$$
31 $$1 + (2.54 + 4.40i)T + (-15.5 + 26.8i)T^{2}$$
37 $$1 + (-2.01 + 3.49i)T + (-18.5 - 32.0i)T^{2}$$
41 $$1 + 5.54T + 41T^{2}$$
43 $$1 + 4.03T + 43T^{2}$$
47 $$1 + (4.64 - 8.04i)T + (-23.5 - 40.7i)T^{2}$$
53 $$1 + (-2.74 - 4.75i)T + (-26.5 + 45.8i)T^{2}$$
59 $$1 + (4.76 + 8.25i)T + (-29.5 + 51.0i)T^{2}$$
61 $$1 + (-0.829 + 1.43i)T + (-30.5 - 52.8i)T^{2}$$
67 $$1 + (1.77 + 3.06i)T + (-33.5 + 58.0i)T^{2}$$
71 $$1 + 2.54T + 71T^{2}$$
73 $$1 + (-4.29 - 7.43i)T + (-36.5 + 63.2i)T^{2}$$
79 $$1 + (6.11 - 10.5i)T + (-39.5 - 68.4i)T^{2}$$
83 $$1 - 14.5T + 83T^{2}$$
89 $$1 + (3.25 - 5.63i)T + (-44.5 - 77.0i)T^{2}$$
97 $$1 + 0.0872T + 97T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}