# Properties

 Degree 2 Conductor $2^{4} \cdot 3^{3} \cdot 7$ Sign $-0.654 - 0.755i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 3·5-s + (−2 + 1.73i)7-s + 5.19i·11-s − 3.46i·13-s − 6·17-s + 1.73i·19-s − 5.19i·23-s + 4·25-s + 10.3i·29-s + 5.19i·31-s + (−6 + 5.19i)35-s + 37-s − 3·41-s − 10·43-s + 6·47-s + ⋯
 L(s)  = 1 + 1.34·5-s + (−0.755 + 0.654i)7-s + 1.56i·11-s − 0.960i·13-s − 1.45·17-s + 0.397i·19-s − 1.08i·23-s + 0.800·25-s + 1.92i·29-s + 0.933i·31-s + (−1.01 + 0.878i)35-s + 0.164·37-s − 0.468·41-s − 1.52·43-s + 0.875·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 - 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$3024$$    =    $$2^{4} \cdot 3^{3} \cdot 7$$ $$\varepsilon$$ = $-0.654 - 0.755i$ motivic weight = $$1$$ character : $\chi_{3024} (1889, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 3024,\ (\ :1/2),\ -0.654 - 0.755i)$$ $$L(1)$$ $$\approx$$ $$1.241958986$$ $$L(\frac12)$$ $$\approx$$ $$1.241958986$$ $$L(\frac{3}{2})$$ 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 + (2 - 1.73i)T$$
good5 $$1 - 3T + 5T^{2}$$
11 $$1 - 5.19iT - 11T^{2}$$
13 $$1 + 3.46iT - 13T^{2}$$
17 $$1 + 6T + 17T^{2}$$
19 $$1 - 1.73iT - 19T^{2}$$
23 $$1 + 5.19iT - 23T^{2}$$
29 $$1 - 10.3iT - 29T^{2}$$
31 $$1 - 5.19iT - 31T^{2}$$
37 $$1 - T + 37T^{2}$$
41 $$1 + 3T + 41T^{2}$$
43 $$1 + 10T + 43T^{2}$$
47 $$1 - 6T + 47T^{2}$$
53 $$1 - 53T^{2}$$
59 $$1 + 6T + 59T^{2}$$
61 $$1 - 13.8iT - 61T^{2}$$
67 $$1 + 2T + 67T^{2}$$
71 $$1 + 5.19iT - 71T^{2}$$
73 $$1 - 3.46iT - 73T^{2}$$
79 $$1 + 14T + 79T^{2}$$
83 $$1 + 6T + 83T^{2}$$
89 $$1 - 9T + 89T^{2}$$
97 $$1 - 6.92iT - 97T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}