# Properties

 Degree 2 Conductor $2^{8} \cdot 3$ Sign $-0.707 + 0.707i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 1

# Related objects

## Dirichlet series

 L(s)  = 1 + i·3-s + 2i·5-s − 4·7-s − 9-s − 4i·11-s + 2i·13-s − 2·15-s − 6·17-s − 4i·19-s − 4i·21-s + 25-s − i·27-s − 2i·29-s − 4·31-s + 4·33-s + ⋯
 L(s)  = 1 + 0.577i·3-s + 0.894i·5-s − 1.51·7-s − 0.333·9-s − 1.20i·11-s + 0.554i·13-s − 0.516·15-s − 1.45·17-s − 0.917i·19-s − 0.872i·21-s + 0.200·25-s − 0.192i·27-s − 0.371i·29-s − 0.718·31-s + 0.696·33-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$768$$    =    $$2^{8} \cdot 3$$ $$\varepsilon$$ = $-0.707 + 0.707i$ motivic weight = $$1$$ character : $\chi_{768} (385, \cdot )$ primitive : yes self-dual : no analytic rank = $$1$$ Selberg data = $$(2,\ 768,\ (\ :1/2),\ -0.707 + 0.707i)$$ $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
3 $$1 - iT$$
good5 $$1 - 2iT - 5T^{2}$$
7 $$1 + 4T + 7T^{2}$$
11 $$1 + 4iT - 11T^{2}$$
13 $$1 - 2iT - 13T^{2}$$
17 $$1 + 6T + 17T^{2}$$
19 $$1 + 4iT - 19T^{2}$$
23 $$1 + 23T^{2}$$
29 $$1 + 2iT - 29T^{2}$$
31 $$1 + 4T + 31T^{2}$$
37 $$1 + 2iT - 37T^{2}$$
41 $$1 + 2T + 41T^{2}$$
43 $$1 + 4iT - 43T^{2}$$
47 $$1 + 8T + 47T^{2}$$
53 $$1 - 10iT - 53T^{2}$$
59 $$1 - 4iT - 59T^{2}$$
61 $$1 + 6iT - 61T^{2}$$
67 $$1 - 4iT - 67T^{2}$$
71 $$1 + 16T + 71T^{2}$$
73 $$1 - 6T + 73T^{2}$$
79 $$1 + 4T + 79T^{2}$$
83 $$1 - 12iT - 83T^{2}$$
89 $$1 + 10T + 89T^{2}$$
97 $$1 + 14T + 97T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}