# Properties

 Degree 2 Conductor $2^{6} \cdot 3^{2} \cdot 7$ Sign $-0.346 + 0.938i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 3.09·5-s − i·7-s + 0.646i·11-s − 4.37i·13-s − 0.295i·17-s − 5.29·19-s − 3.94·23-s + 4.58·25-s − 5.03·29-s − 9.16i·31-s − 3.09i·35-s − 2.55i·37-s − 10.4i·41-s − 1.63·43-s − 5.65·47-s + ⋯
 L(s)  = 1 + 1.38·5-s − 0.377i·7-s + 0.194i·11-s − 1.21i·13-s − 0.0715i·17-s − 1.21·19-s − 0.823·23-s + 0.916·25-s − 0.934·29-s − 1.64i·31-s − 0.523i·35-s − 0.419i·37-s − 1.62i·41-s − 0.249·43-s − 0.825·47-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.346 + 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.346 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$4032$$    =    $$2^{6} \cdot 3^{2} \cdot 7$$ $$\varepsilon$$ = $-0.346 + 0.938i$ motivic weight = $$1$$ character : $\chi_{4032} (2591, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 4032,\ (\ :1/2),\ -0.346 + 0.938i)$ $L(1)$ $\approx$ $1.684283526$ $L(\frac12)$ $\approx$ $1.684283526$ $L(\frac{3}{2})$ 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 + iT$$
good5 $$1 - 3.09T + 5T^{2}$$
11 $$1 - 0.646iT - 11T^{2}$$
13 $$1 + 4.37iT - 13T^{2}$$
17 $$1 + 0.295iT - 17T^{2}$$
19 $$1 + 5.29T + 19T^{2}$$
23 $$1 + 3.94T + 23T^{2}$$
29 $$1 + 5.03T + 29T^{2}$$
31 $$1 + 9.16iT - 31T^{2}$$
37 $$1 + 2.55iT - 37T^{2}$$
41 $$1 + 10.4iT - 41T^{2}$$
43 $$1 + 1.63T + 43T^{2}$$
47 $$1 + 5.65T + 47T^{2}$$
53 $$1 - 8.64T + 53T^{2}$$
59 $$1 - 59T^{2}$$
61 $$1 + 6.92iT - 61T^{2}$$
67 $$1 + 2.55T + 67T^{2}$$
71 $$1 - 1.11T + 71T^{2}$$
73 $$1 + 9.58T + 73T^{2}$$
79 $$1 - 16.7iT - 79T^{2}$$
83 $$1 + 8.50iT - 83T^{2}$$
89 $$1 - 10.4iT - 89T^{2}$$
97 $$1 + 2.41T + 97T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}