# Properties

 Degree 2 Conductor $3^{2} \cdot 11 \cdot 19^{2}$ Sign $-1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 1

# Related objects

## Dirichlet series

 L(s)  = 1 + 2-s − 4-s + 2·5-s + 4·7-s − 3·8-s + 2·10-s − 11-s + 2·13-s + 4·14-s − 16-s + 2·17-s − 2·20-s − 22-s − 8·23-s − 25-s + 2·26-s − 4·28-s − 6·29-s + 8·31-s + 5·32-s + 2·34-s + 8·35-s − 6·37-s − 6·40-s − 2·41-s + 44-s − 8·46-s + ⋯
 L(s)  = 1 + 0.707·2-s − 1/2·4-s + 0.894·5-s + 1.51·7-s − 1.06·8-s + 0.632·10-s − 0.301·11-s + 0.554·13-s + 1.06·14-s − 1/4·16-s + 0.485·17-s − 0.447·20-s − 0.213·22-s − 1.66·23-s − 1/5·25-s + 0.392·26-s − 0.755·28-s − 1.11·29-s + 1.43·31-s + 0.883·32-s + 0.342·34-s + 1.35·35-s − 0.986·37-s − 0.948·40-s − 0.312·41-s + 0.150·44-s − 1.17·46-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 35739 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 35739 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$35739$$    =    $$3^{2} \cdot 11 \cdot 19^{2}$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{35739} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 1 Selberg data = $(2,\ 35739,\ (\ :1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{3,\;11,\;19\}$, $F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{3,\;11,\;19\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 $$1$$
11 $$1 + T$$
19 $$1$$
good2 $$1 - T + p T^{2}$$
5 $$1 - 2 T + p T^{2}$$
7 $$1 - 4 T + p T^{2}$$
13 $$1 - 2 T + p T^{2}$$
17 $$1 - 2 T + p T^{2}$$
23 $$1 + 8 T + p T^{2}$$
29 $$1 + 6 T + p T^{2}$$
31 $$1 - 8 T + p T^{2}$$
37 $$1 + 6 T + p T^{2}$$
41 $$1 + 2 T + p T^{2}$$
43 $$1 + p T^{2}$$
47 $$1 + 8 T + p T^{2}$$
53 $$1 - 6 T + p T^{2}$$
59 $$1 + 4 T + p T^{2}$$
61 $$1 - 6 T + p T^{2}$$
67 $$1 - 4 T + p T^{2}$$
71 $$1 + p T^{2}$$
73 $$1 + 14 T + p T^{2}$$
79 $$1 - 4 T + p T^{2}$$
83 $$1 + 12 T + p T^{2}$$
89 $$1 + 6 T + p T^{2}$$
97 $$1 + 2 T + p T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}