# Properties

 Degree 2 Conductor $2^{6} \cdot 431$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 3·3-s − 3·5-s + 4·7-s + 6·9-s + 5·11-s − 4·13-s − 9·15-s − 6·17-s + 7·19-s + 12·21-s − 3·23-s + 4·25-s + 9·27-s − 3·29-s + 4·31-s + 15·33-s − 12·35-s − 8·37-s − 12·39-s − 6·41-s + 8·43-s − 18·45-s + 6·47-s + 9·49-s − 18·51-s − 53-s − 15·55-s + ⋯
 L(s)  = 1 + 1.73·3-s − 1.34·5-s + 1.51·7-s + 2·9-s + 1.50·11-s − 1.10·13-s − 2.32·15-s − 1.45·17-s + 1.60·19-s + 2.61·21-s − 0.625·23-s + 4/5·25-s + 1.73·27-s − 0.557·29-s + 0.718·31-s + 2.61·33-s − 2.02·35-s − 1.31·37-s − 1.92·39-s − 0.937·41-s + 1.21·43-s − 2.68·45-s + 0.875·47-s + 9/7·49-s − 2.52·51-s − 0.137·53-s − 2.02·55-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 27584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\n
\begin{aligned} \Lambda(s)=\mathstrut & 27584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\n

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$27584$$    =    $$2^{6} \cdot 431$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{27584} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 27584,\ (\ :1/2),\ 1)$ $L(1)$ $\approx$ $4.733567115$ $L(\frac12)$ $\approx$ $4.733567115$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;431\}$, $F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{2,\;431\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
431 $$1 + T$$
good3 $$1 - p T + p T^{2}$$
5 $$1 + 3 T + p T^{2}$$
7 $$1 - 4 T + p T^{2}$$
11 $$1 - 5 T + p T^{2}$$
13 $$1 + 4 T + p T^{2}$$
17 $$1 + 6 T + p T^{2}$$
19 $$1 - 7 T + p T^{2}$$
23 $$1 + 3 T + p T^{2}$$
29 $$1 + 3 T + p T^{2}$$
31 $$1 - 4 T + p T^{2}$$
37 $$1 + 8 T + p T^{2}$$
41 $$1 + 6 T + p T^{2}$$
43 $$1 - 8 T + p T^{2}$$
47 $$1 - 6 T + p T^{2}$$
53 $$1 + T + p T^{2}$$
59 $$1 - 9 T + p T^{2}$$
61 $$1 + 2 T + p T^{2}$$
67 $$1 + 10 T + p T^{2}$$
71 $$1 - 8 T + p T^{2}$$
73 $$1 + 6 T + p T^{2}$$
79 $$1 - 16 T + p T^{2}$$
83 $$1 - 16 T + p T^{2}$$
89 $$1 + 6 T + p T^{2}$$
97 $$1 + 5 T + p T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}