# Properties

 Degree 2 Conductor $2^{6} \cdot 3^{2} \cdot 7$ Sign $0.258 + 0.965i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 0.732i·5-s − 7-s + 1.46i·11-s − 3.26i·13-s − 2·17-s − 4.73i·19-s + 3.46·23-s + 4.46·25-s + 5.46i·29-s − 4·31-s − 0.732i·35-s + 5.46i·37-s + 2·41-s − 1.46i·43-s − 10.9·47-s + ⋯
 L(s)  = 1 + 0.327i·5-s − 0.377·7-s + 0.441i·11-s − 0.906i·13-s − 0.485·17-s − 1.08i·19-s + 0.722·23-s + 0.892·25-s + 1.01i·29-s − 0.718·31-s − 0.123i·35-s + 0.898i·37-s + 0.312·41-s − 0.223i·43-s − 1.59·47-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$4032$$    =    $$2^{6} \cdot 3^{2} \cdot 7$$ $$\varepsilon$$ = $0.258 + 0.965i$ motivic weight = $$1$$ character : $\chi_{4032} (2017, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 4032,\ (\ :1/2),\ 0.258 + 0.965i)$ $L(1)$ $\approx$ $1.279569950$ $L(\frac12)$ $\approx$ $1.279569950$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3,\;7\}$, $$F_p$$ is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 $$1$$
3 $$1$$
7 $$1 + T$$
good5 $$1 - 0.732iT - 5T^{2}$$
11 $$1 - 1.46iT - 11T^{2}$$
13 $$1 + 3.26iT - 13T^{2}$$
17 $$1 + 2T + 17T^{2}$$
19 $$1 + 4.73iT - 19T^{2}$$
23 $$1 - 3.46T + 23T^{2}$$
29 $$1 - 5.46iT - 29T^{2}$$
31 $$1 + 4T + 31T^{2}$$
37 $$1 - 5.46iT - 37T^{2}$$
41 $$1 - 2T + 41T^{2}$$
43 $$1 + 1.46iT - 43T^{2}$$
47 $$1 + 10.9T + 47T^{2}$$
53 $$1 + 12iT - 53T^{2}$$
59 $$1 + 7.66iT - 59T^{2}$$
61 $$1 + 13.1iT - 61T^{2}$$
67 $$1 - 8iT - 67T^{2}$$
71 $$1 + 10.9T + 71T^{2}$$
73 $$1 + 0.928T + 73T^{2}$$
79 $$1 + 2.92T + 79T^{2}$$
83 $$1 + 11.6iT - 83T^{2}$$
89 $$1 - 15.8T + 89T^{2}$$
97 $$1 + 4.92T + 97T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}