# Properties

 Degree 2 Conductor $2^{5} \cdot 3^{3} \cdot 7$ Sign $0.971 - 0.236i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 4.18·5-s − i·7-s − 1.86i·11-s + 3.07i·13-s − 0.504i·17-s − 3.05·19-s − 5.97·23-s + 12.4·25-s − 3.52·29-s − 10.8i·31-s + 4.18i·35-s + 11.3i·37-s + 7.26i·41-s − 9.02·43-s + 0.327·47-s + ⋯
 L(s)  = 1 − 1.87·5-s − 0.377i·7-s − 0.563i·11-s + 0.853i·13-s − 0.122i·17-s − 0.701·19-s − 1.24·23-s + 2.49·25-s − 0.655·29-s − 1.95i·31-s + 0.706i·35-s + 1.86i·37-s + 1.13i·41-s − 1.37·43-s + 0.0477·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.236i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.971 - 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$6048$$    =    $$2^{5} \cdot 3^{3} \cdot 7$$ $$\varepsilon$$ = $0.971 - 0.236i$ motivic weight = $$1$$ character : $\chi_{6048} (5615, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 6048,\ (\ :1/2),\ 0.971 - 0.236i)$ $L(1)$ $\approx$ $0.6463040688$ $L(\frac12)$ $\approx$ $0.6463040688$ $L(\frac{3}{2})$ 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 + iT$$
good5 $$1 + 4.18T + 5T^{2}$$
11 $$1 + 1.86iT - 11T^{2}$$
13 $$1 - 3.07iT - 13T^{2}$$
17 $$1 + 0.504iT - 17T^{2}$$
19 $$1 + 3.05T + 19T^{2}$$
23 $$1 + 5.97T + 23T^{2}$$
29 $$1 + 3.52T + 29T^{2}$$
31 $$1 + 10.8iT - 31T^{2}$$
37 $$1 - 11.3iT - 37T^{2}$$
41 $$1 - 7.26iT - 41T^{2}$$
43 $$1 + 9.02T + 43T^{2}$$
47 $$1 - 0.327T + 47T^{2}$$
53 $$1 + 8.32T + 53T^{2}$$
59 $$1 + 5.98iT - 59T^{2}$$
61 $$1 + 13.2iT - 61T^{2}$$
67 $$1 - 9.21T + 67T^{2}$$
71 $$1 + 9.02T + 71T^{2}$$
73 $$1 - 0.416T + 73T^{2}$$
79 $$1 - 6.00iT - 79T^{2}$$
83 $$1 - 2.62iT - 83T^{2}$$
89 $$1 + 5.69iT - 89T^{2}$$
97 $$1 - 6.21T + 97T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}