# Properties

 Degree 2 Conductor $2^{8} \cdot 3^{2}$ Sign $0.707 - 0.707i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 2i·5-s − 4·7-s − 4i·11-s − 2i·13-s + 6·17-s + 4i·19-s + 25-s − 2i·29-s − 4·31-s − 8i·35-s + 2i·37-s + 2·41-s + 4i·43-s + 8·47-s + 9·49-s + ⋯
 L(s)  = 1 + 0.894i·5-s − 1.51·7-s − 1.20i·11-s − 0.554i·13-s + 1.45·17-s + 0.917i·19-s + 0.200·25-s − 0.371i·29-s − 0.718·31-s − 1.35i·35-s + 0.328i·37-s + 0.312·41-s + 0.609i·43-s + 1.16·47-s + 1.28·49-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$2304$$    =    $$2^{8} \cdot 3^{2}$$ $$\varepsilon$$ = $0.707 - 0.707i$ motivic weight = $$1$$ character : $\chi_{2304} (1153, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 2304,\ (\ :1/2),\ 0.707 - 0.707i)$$ $$L(1)$$ $$\approx$$ $$1.352639401$$ $$L(\frac12)$$ $$\approx$$ $$1.352639401$$ $$L(\frac{3}{2})$$ 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$$
good5 $$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}