# Properties

 Degree 2 Conductor $2^{5} \cdot 3^{3} \cdot 7$ Sign $0.707 - 0.707i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 1.19i·5-s − i·7-s + 1.62·11-s − 2.09·13-s − 0.585i·17-s − 5.89i·19-s − 7.09·23-s + 3.56·25-s + 7.08i·29-s + 6.72i·31-s + 1.19·35-s + 2.17·37-s + 0.164i·41-s + 5.75i·43-s + 3.41·47-s + ⋯
 L(s)  = 1 + 0.536i·5-s − 0.377i·7-s + 0.491·11-s − 0.581·13-s − 0.142i·17-s − 1.35i·19-s − 1.47·23-s + 0.712·25-s + 1.31i·29-s + 1.20i·31-s + 0.202·35-s + 0.357·37-s + 0.0256i·41-s + 0.877i·43-s + 0.498·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$6048$$    =    $$2^{5} \cdot 3^{3} \cdot 7$$ $$\varepsilon$$ = $0.707 - 0.707i$ motivic weight = $$1$$ character : $\chi_{6048} (2591, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 6048,\ (\ :1/2),\ 0.707 - 0.707i)$ $L(1)$ $\approx$ $1.710053417$ $L(\frac12)$ $\approx$ $1.710053417$ $L(\frac{3}{2})$ 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 - 1.19iT - 5T^{2}$$
11 $$1 - 1.62T + 11T^{2}$$
13 $$1 + 2.09T + 13T^{2}$$
17 $$1 + 0.585iT - 17T^{2}$$
19 $$1 + 5.89iT - 19T^{2}$$
23 $$1 + 7.09T + 23T^{2}$$
29 $$1 - 7.08iT - 29T^{2}$$
31 $$1 - 6.72iT - 31T^{2}$$
37 $$1 - 2.17T + 37T^{2}$$
41 $$1 - 0.164iT - 41T^{2}$$
43 $$1 - 5.75iT - 43T^{2}$$
47 $$1 - 3.41T + 47T^{2}$$
53 $$1 - 3.29iT - 53T^{2}$$
59 $$1 + 2.08T + 59T^{2}$$
61 $$1 - 4.92T + 61T^{2}$$
67 $$1 + 12.4iT - 67T^{2}$$
71 $$1 - 11.2T + 71T^{2}$$
73 $$1 - 3.97T + 73T^{2}$$
79 $$1 + 0.702iT - 79T^{2}$$
83 $$1 - 16.8T + 83T^{2}$$
89 $$1 - 6.33iT - 89T^{2}$$
97 $$1 + 6.58T + 97T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}