# Properties

 Degree 2 Conductor $2^{6} \cdot 3^{2} \cdot 7$ Sign $0.169 - 0.985i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + i·7-s − 4.24i·11-s + 6i·13-s + 4.24·23-s − 5·25-s + 4.24·29-s + 4i·31-s − 6i·37-s + 8.48i·41-s − 6·43-s + 8.48·47-s − 49-s − 12.7·53-s + 8.48i·59-s − 6i·61-s + ⋯
 L(s)  = 1 + 0.377i·7-s − 1.27i·11-s + 1.66i·13-s + 0.884·23-s − 25-s + 0.787·29-s + 0.718i·31-s − 0.986i·37-s + 1.32i·41-s − 0.914·43-s + 1.23·47-s − 0.142·49-s − 1.74·53-s + 1.10i·59-s − 0.768i·61-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.169 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$4032$$    =    $$2^{6} \cdot 3^{2} \cdot 7$$ $$\varepsilon$$ = $0.169 - 0.985i$ motivic weight = $$1$$ character : $\chi_{4032} (2591, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 4032,\ (\ :1/2),\ 0.169 - 0.985i)$ $L(1)$ $\approx$ $1.495257923$ $L(\frac12)$ $\approx$ $1.495257923$ $L(\frac{3}{2})$ 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 + 5T^{2}$$
11 $$1 + 4.24iT - 11T^{2}$$
13 $$1 - 6iT - 13T^{2}$$
17 $$1 - 17T^{2}$$
19 $$1 + 19T^{2}$$
23 $$1 - 4.24T + 23T^{2}$$
29 $$1 - 4.24T + 29T^{2}$$
31 $$1 - 4iT - 31T^{2}$$
37 $$1 + 6iT - 37T^{2}$$
41 $$1 - 8.48iT - 41T^{2}$$
43 $$1 + 6T + 43T^{2}$$
47 $$1 - 8.48T + 47T^{2}$$
53 $$1 + 12.7T + 53T^{2}$$
59 $$1 - 8.48iT - 59T^{2}$$
61 $$1 + 6iT - 61T^{2}$$
67 $$1 - 12T + 67T^{2}$$
71 $$1 - 4.24T + 71T^{2}$$
73 $$1 + 2T + 73T^{2}$$
79 $$1 - 10iT - 79T^{2}$$
83 $$1 - 16.9iT - 83T^{2}$$
89 $$1 - 8.48iT - 89T^{2}$$
97 $$1 + 10T + 97T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}