# Properties

 Degree 2 Conductor $2^{6} \cdot 3^{3}$ Sign $-1$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 2.23i·5-s − 3.87i·7-s − 1.73·11-s − 2·13-s − 4.47i·17-s − 6.92·23-s + 4.47i·29-s + 3.87i·31-s − 8.66·35-s + 4·37-s + 8.94i·41-s − 7.74i·43-s − 3.46·47-s − 8.00·49-s − 2.23i·53-s + ⋯
 L(s)  = 1 − 0.999i·5-s − 1.46i·7-s − 0.522·11-s − 0.554·13-s − 1.08i·17-s − 1.44·23-s + 0.830i·29-s + 0.695i·31-s − 1.46·35-s + 0.657·37-s + 1.39i·41-s − 1.18i·43-s − 0.505·47-s − 1.14·49-s − 0.307i·53-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$1728$$    =    $$2^{6} \cdot 3^{3}$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{1728} (1727, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 1728,\ (\ :1/2),\ -1)$$ $$L(1)$$ $$\approx$$ $$0.8379584553$$ $$L(\frac12)$$ $$\approx$$ $$0.8379584553$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
good5 $$1 + 2.23iT - 5T^{2}$$
7 $$1 + 3.87iT - 7T^{2}$$
11 $$1 + 1.73T + 11T^{2}$$
13 $$1 + 2T + 13T^{2}$$
17 $$1 + 4.47iT - 17T^{2}$$
19 $$1 - 19T^{2}$$
23 $$1 + 6.92T + 23T^{2}$$
29 $$1 - 4.47iT - 29T^{2}$$
31 $$1 - 3.87iT - 31T^{2}$$
37 $$1 - 4T + 37T^{2}$$
41 $$1 - 8.94iT - 41T^{2}$$
43 $$1 + 7.74iT - 43T^{2}$$
47 $$1 + 3.46T + 47T^{2}$$
53 $$1 + 2.23iT - 53T^{2}$$
59 $$1 + 3.46T + 59T^{2}$$
61 $$1 - 4T + 61T^{2}$$
67 $$1 - 7.74iT - 67T^{2}$$
71 $$1 + 10.3T + 71T^{2}$$
73 $$1 - 5T + 73T^{2}$$
79 $$1 - 7.74iT - 79T^{2}$$
83 $$1 + 12.1T + 83T^{2}$$
89 $$1 + 4.47iT - 89T^{2}$$
97 $$1 - 11T + 97T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}