# Properties

 Degree 2 Conductor $3 \cdot 211$ Sign $0.751 - 0.660i$ Motivic weight 0 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.587 − 0.190i)2-s + (0.809 + 0.587i)3-s + (−0.5 + 0.363i)4-s + (0.587 + 0.809i)5-s + (0.587 + 0.190i)6-s + (0.309 − 0.951i)7-s + (−0.587 + 0.809i)8-s + (0.309 + 0.951i)9-s + (0.5 + 0.363i)10-s + (−0.951 − 1.30i)11-s − 0.618·12-s + (−0.309 + 0.951i)13-s − 0.618i·14-s + i·15-s + (−1.53 − 0.5i)17-s + (0.363 + 0.5i)18-s + ⋯
 L(s)  = 1 + (0.587 − 0.190i)2-s + (0.809 + 0.587i)3-s + (−0.5 + 0.363i)4-s + (0.587 + 0.809i)5-s + (0.587 + 0.190i)6-s + (0.309 − 0.951i)7-s + (−0.587 + 0.809i)8-s + (0.309 + 0.951i)9-s + (0.5 + 0.363i)10-s + (−0.951 − 1.30i)11-s − 0.618·12-s + (−0.309 + 0.951i)13-s − 0.618i·14-s + i·15-s + (−1.53 − 0.5i)17-s + (0.363 + 0.5i)18-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 633 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 - 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 633 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 - 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$633$$    =    $$3 \cdot 211$$ $$\varepsilon$$ = $0.751 - 0.660i$ motivic weight = $$0$$ character : $\chi_{633} (266, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 633,\ (\ :0),\ 0.751 - 0.660i)$$ $$L(\frac{1}{2})$$ $$\approx$$ $$1.424903184$$ $$L(\frac12)$$ $$\approx$$ $$1.424903184$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{3,\;211\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{3,\;211\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 $$1 + (-0.809 - 0.587i)T$$
211 $$1 + (-0.309 + 0.951i)T$$
good2 $$1 + (-0.587 + 0.190i)T + (0.809 - 0.587i)T^{2}$$
5 $$1 + (-0.587 - 0.809i)T + (-0.309 + 0.951i)T^{2}$$
7 $$1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2}$$
11 $$1 + (0.951 + 1.30i)T + (-0.309 + 0.951i)T^{2}$$
13 $$1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2}$$
17 $$1 + (1.53 + 0.5i)T + (0.809 + 0.587i)T^{2}$$
19 $$1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2}$$
23 $$1 + (-0.951 + 0.309i)T + (0.809 - 0.587i)T^{2}$$
29 $$1 + (-0.587 - 0.190i)T + (0.809 + 0.587i)T^{2}$$
31 $$1 - 0.618T + T^{2}$$
37 $$1 + (0.309 + 0.951i)T^{2}$$
41 $$1 + (-0.951 - 1.30i)T + (-0.309 + 0.951i)T^{2}$$
43 $$1 + T + T^{2}$$
47 $$1 + (0.587 + 0.190i)T + (0.809 + 0.587i)T^{2}$$
53 $$1 + (-0.309 - 0.951i)T^{2}$$
59 $$1 + (-0.309 + 0.951i)T^{2}$$
61 $$1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2}$$
67 $$1 + T^{2}$$
71 $$1 + (0.363 - 0.5i)T + (-0.309 - 0.951i)T^{2}$$
73 $$1 + T + T^{2}$$
79 $$1 + (0.309 - 0.951i)T^{2}$$
83 $$1 + (0.363 + 0.5i)T + (-0.309 + 0.951i)T^{2}$$
89 $$1 + (0.951 + 0.309i)T + (0.809 + 0.587i)T^{2}$$
97 $$1 + (0.309 + 0.951i)T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}