# Properties

 Degree 2 Conductor $3^{2} \cdot 5 \cdot 7 \cdot 17^{2}$ Sign $-1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 1

# Related objects

## Dirichlet series

 L(s)  = 1 − 2-s − 4-s + 5-s − 7-s + 3·8-s − 10-s − 6·13-s + 14-s − 16-s − 8·19-s − 20-s + 8·23-s + 25-s + 6·26-s + 28-s − 2·29-s − 4·31-s − 5·32-s − 35-s + 2·37-s + 8·38-s + 3·40-s − 6·41-s + 4·43-s − 8·46-s − 8·47-s + 49-s + ⋯
 L(s)  = 1 − 0.707·2-s − 1/2·4-s + 0.447·5-s − 0.377·7-s + 1.06·8-s − 0.316·10-s − 1.66·13-s + 0.267·14-s − 1/4·16-s − 1.83·19-s − 0.223·20-s + 1.66·23-s + 1/5·25-s + 1.17·26-s + 0.188·28-s − 0.371·29-s − 0.718·31-s − 0.883·32-s − 0.169·35-s + 0.328·37-s + 1.29·38-s + 0.474·40-s − 0.937·41-s + 0.609·43-s − 1.17·46-s − 1.16·47-s + 1/7·49-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 91035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\n
\begin{aligned} \Lambda(s)=\mathstrut & 91035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\n

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$91035$$    =    $$3^{2} \cdot 5 \cdot 7 \cdot 17^{2}$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{91035} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 1 Selberg data = $(2,\ 91035,\ (\ :1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{3,\;5,\;7,\;17\}$, $F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{3,\;5,\;7,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 $$1$$
5 $$1 - T$$
7 $$1 + T$$
17 $$1$$
good2 $$1 + T + p T^{2}$$
11 $$1 + p T^{2}$$
13 $$1 + 6 T + p T^{2}$$
19 $$1 + 8 T + p T^{2}$$
23 $$1 - 8 T + p T^{2}$$
29 $$1 + 2 T + p T^{2}$$
31 $$1 + 4 T + p T^{2}$$
37 $$1 - 2 T + p T^{2}$$
41 $$1 + 6 T + p T^{2}$$
43 $$1 - 4 T + p T^{2}$$
47 $$1 + 8 T + p T^{2}$$
53 $$1 + 10 T + p T^{2}$$
59 $$1 + 4 T + p T^{2}$$
61 $$1 - 2 T + p T^{2}$$
67 $$1 - 4 T + p T^{2}$$
71 $$1 + 12 T + p T^{2}$$
73 $$1 - 2 T + p T^{2}$$
79 $$1 + 8 T + p T^{2}$$
83 $$1 - 4 T + p T^{2}$$
89 $$1 - 6 T + p T^{2}$$
97 $$1 - 18 T + p T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}