# Properties

 Degree 2 Conductor $2^{4} \cdot 7 \cdot 19 \cdot 47$ Sign $-1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 1

# Related objects

## Dirichlet series

 L(s)  = 1 − 3-s + 5-s + 7-s − 2·9-s + 2·11-s + 4·13-s − 15-s − 6·17-s − 19-s − 21-s − 6·23-s − 4·25-s + 5·27-s − 7·29-s − 2·31-s − 2·33-s + 35-s + 2·37-s − 4·39-s + 6·41-s + 6·43-s − 2·45-s + 47-s + 49-s + 6·51-s + 2·55-s + 57-s + ⋯
 L(s)  = 1 − 0.577·3-s + 0.447·5-s + 0.377·7-s − 2/3·9-s + 0.603·11-s + 1.10·13-s − 0.258·15-s − 1.45·17-s − 0.229·19-s − 0.218·21-s − 1.25·23-s − 4/5·25-s + 0.962·27-s − 1.29·29-s − 0.359·31-s − 0.348·33-s + 0.169·35-s + 0.328·37-s − 0.640·39-s + 0.937·41-s + 0.914·43-s − 0.298·45-s + 0.145·47-s + 1/7·49-s + 0.840·51-s + 0.269·55-s + 0.132·57-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 100016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 100016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$100016$$    =    $$2^{4} \cdot 7 \cdot 19 \cdot 47$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{100016} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 1 Selberg data = $(2,\ 100016,\ (\ :1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;7,\;19,\;47\}$, $F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{2,\;7,\;19,\;47\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 $$1$$
7 $$1 - T$$
19 $$1 + T$$
47 $$1 - T$$
good3 $$1 + T + p T^{2}$$
5 $$1 - T + p T^{2}$$
11 $$1 - 2 T + p T^{2}$$
13 $$1 - 4 T + p T^{2}$$
17 $$1 + 6 T + p T^{2}$$
23 $$1 + 6 T + p T^{2}$$
29 $$1 + 7 T + p T^{2}$$
31 $$1 + 2 T + p T^{2}$$
37 $$1 - 2 T + p T^{2}$$
41 $$1 - 6 T + p T^{2}$$
43 $$1 - 6 T + p T^{2}$$
53 $$1 + p T^{2}$$
59 $$1 + 12 T + p T^{2}$$
61 $$1 + p T^{2}$$
67 $$1 - 5 T + p T^{2}$$
71 $$1 + 6 T + p T^{2}$$
73 $$1 + 11 T + p T^{2}$$
79 $$1 + 2 T + p T^{2}$$
83 $$1 - 6 T + p T^{2}$$
89 $$1 - 9 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}