# Properties

 Degree 2 Conductor $7 \cdot 19$ Sign $0.305 - 0.952i$ Motivic weight 0 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.5 + 0.866i)2-s + (−0.866 − 0.5i)3-s + (0.866 + 0.5i)5-s + (0.866 − 0.499i)6-s + i·7-s − 8-s + (−0.866 + 0.499i)10-s + (0.866 − 0.5i)13-s + (−0.866 − 0.5i)14-s + (−0.499 − 0.866i)15-s + (0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s − i·19-s + (0.5 − 0.866i)21-s + (−0.5 − 0.866i)23-s + (0.866 + 0.499i)24-s + ⋯
 L(s)  = 1 + (−0.5 + 0.866i)2-s + (−0.866 − 0.5i)3-s + (0.866 + 0.5i)5-s + (0.866 − 0.499i)6-s + i·7-s − 8-s + (−0.866 + 0.499i)10-s + (0.866 − 0.5i)13-s + (−0.866 − 0.5i)14-s + (−0.499 − 0.866i)15-s + (0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s − i·19-s + (0.5 − 0.866i)21-s + (−0.5 − 0.866i)23-s + (0.866 + 0.499i)24-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$133$$    =    $$7 \cdot 19$$ $$\varepsilon$$ = $0.305 - 0.952i$ motivic weight = $$0$$ character : $\chi_{133} (125, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 133,\ (\ :0),\ 0.305 - 0.952i)$$ $$L(\frac{1}{2})$$ $$\approx$$ $$0.4405601059$$ $$L(\frac12)$$ $$\approx$$ $$0.4405601059$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{7,\;19\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{7,\;19\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad7 $$1 - iT$$
19 $$1 + iT$$
good2 $$1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}$$
3 $$1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}$$
5 $$1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}$$
11 $$1 + T^{2}$$
13 $$1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}$$
17 $$1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}$$
23 $$1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}$$
29 $$1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}$$
31 $$1 - T^{2}$$
37 $$1 + T^{2}$$
41 $$1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}$$
43 $$1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}$$
47 $$1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}$$
53 $$1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}$$
59 $$1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}$$
61 $$1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}$$
67 $$1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}$$
71 $$1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}$$
73 $$1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}$$
79 $$1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}$$
83 $$1 - T^{2}$$
89 $$1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}$$
97 $$1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}