# Properties

 Degree 2 Conductor $2^{4} \cdot 3^{3} \cdot 7$ Sign $-0.990 - 0.135i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 3.63i·5-s + (1 + 2.44i)7-s − 1.50i·11-s + 5.91i·13-s + 5.14i·17-s − 5.24·19-s − 8.78i·23-s − 8.24·25-s − 8.91·29-s − 3.24·31-s + (8.91 − 3.63i)35-s + 37-s − 6.65i·41-s − 9.37i·43-s + 3.69·47-s + ⋯
 L(s)  = 1 − 1.62i·5-s + (0.377 + 0.925i)7-s − 0.454i·11-s + 1.64i·13-s + 1.24i·17-s − 1.20·19-s − 1.83i·23-s − 1.64·25-s − 1.65·29-s − 0.582·31-s + (1.50 − 0.615i)35-s + 0.164·37-s − 1.03i·41-s − 1.43i·43-s + 0.538·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.135i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 - 0.135i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$3024$$    =    $$2^{4} \cdot 3^{3} \cdot 7$$ $$\varepsilon$$ = $-0.990 - 0.135i$ motivic weight = $$1$$ character : $\chi_{3024} (1567, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 3024,\ (\ :1/2),\ -0.990 - 0.135i)$$ $$L(1)$$ $$\approx$$ $$0.2348839587$$ $$L(\frac12)$$ $$\approx$$ $$0.2348839587$$ $$L(\frac{3}{2})$$ 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 + (-1 - 2.44i)T$$
good5 $$1 + 3.63iT - 5T^{2}$$
11 $$1 + 1.50iT - 11T^{2}$$
13 $$1 - 5.91iT - 13T^{2}$$
17 $$1 - 5.14iT - 17T^{2}$$
19 $$1 + 5.24T + 19T^{2}$$
23 $$1 + 8.78iT - 23T^{2}$$
29 $$1 + 8.91T + 29T^{2}$$
31 $$1 + 3.24T + 31T^{2}$$
37 $$1 - T + 37T^{2}$$
41 $$1 + 6.65iT - 41T^{2}$$
43 $$1 + 9.37iT - 43T^{2}$$
47 $$1 - 3.69T + 47T^{2}$$
53 $$1 + 12.6T + 53T^{2}$$
59 $$1 + 8.91T + 59T^{2}$$
61 $$1 + 3.46iT - 61T^{2}$$
67 $$1 + 10.8iT - 67T^{2}$$
71 $$1 - 3.63iT - 71T^{2}$$
73 $$1 - 0.420iT - 73T^{2}$$
79 $$1 + 4.47iT - 79T^{2}$$
83 $$1 + 3.69T + 83T^{2}$$
89 $$1 - 13.9iT - 89T^{2}$$
97 $$1 - 13.2iT - 97T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}