# Properties

 Degree 2 Conductor $2^{5} \cdot 3^{3} \cdot 7$ Sign $-0.858 + 0.513i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 3.06i·5-s + 7-s + 5.80i·11-s − 3.52i·13-s − 6.79·17-s − 5.28i·19-s + 5.65·23-s − 4.41·25-s + 1.21i·29-s − 0.107·31-s + 3.06i·35-s − 4.90i·37-s − 11.6·41-s + 1.85i·43-s − 6.76·47-s + ⋯
 L(s)  = 1 + 1.37i·5-s + 0.377·7-s + 1.75i·11-s − 0.977i·13-s − 1.64·17-s − 1.21i·19-s + 1.17·23-s − 0.883·25-s + 0.225i·29-s − 0.0192·31-s + 0.518i·35-s − 0.805i·37-s − 1.82·41-s + 0.283i·43-s − 0.986·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.858 + 0.513i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.858 + 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$6048$$    =    $$2^{5} \cdot 3^{3} \cdot 7$$ $$\varepsilon$$ = $-0.858 + 0.513i$ motivic weight = $$1$$ character : $\chi_{6048} (3025, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 6048,\ (\ :1/2),\ -0.858 + 0.513i)$$ $$L(1)$$ $$\approx$$ $$0.4491313753$$ $$L(\frac12)$$ $$\approx$$ $$0.4491313753$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;7\}$,$$F_p(T)$$ 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 - T$$
good5 $$1 - 3.06iT - 5T^{2}$$
11 $$1 - 5.80iT - 11T^{2}$$
13 $$1 + 3.52iT - 13T^{2}$$
17 $$1 + 6.79T + 17T^{2}$$
19 $$1 + 5.28iT - 19T^{2}$$
23 $$1 - 5.65T + 23T^{2}$$
29 $$1 - 1.21iT - 29T^{2}$$
31 $$1 + 0.107T + 31T^{2}$$
37 $$1 + 4.90iT - 37T^{2}$$
41 $$1 + 11.6T + 41T^{2}$$
43 $$1 - 1.85iT - 43T^{2}$$
47 $$1 + 6.76T + 47T^{2}$$
53 $$1 - 11.4iT - 53T^{2}$$
59 $$1 - 7.85iT - 59T^{2}$$
61 $$1 - 12.0iT - 61T^{2}$$
67 $$1 - 6.66iT - 67T^{2}$$
71 $$1 + 1.98T + 71T^{2}$$
73 $$1 + 6.52T + 73T^{2}$$
79 $$1 + 2.30T + 79T^{2}$$
83 $$1 + 12.5iT - 83T^{2}$$
89 $$1 + 7.79T + 89T^{2}$$
97 $$1 - 4.05T + 97T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}