# Properties

 Degree 2 Conductor $2^{4} \cdot 5^{2} \cdot 7^{2}$ Sign $-1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 1

# Related objects

## Dirichlet series

 L(s)  = 1 − 3·3-s + 6·9-s + 2·13-s + 2·17-s + 2·19-s − 23-s − 9·27-s − 29-s − 10·31-s + 8·37-s − 6·39-s − 3·41-s + 5·43-s + 8·47-s − 6·51-s + 6·53-s − 6·57-s − 2·59-s − 9·61-s − 7·67-s + 3·69-s − 6·71-s + 10·73-s + 10·79-s + 9·81-s − 9·83-s + 3·87-s + ⋯
 L(s)  = 1 − 1.73·3-s + 2·9-s + 0.554·13-s + 0.485·17-s + 0.458·19-s − 0.208·23-s − 1.73·27-s − 0.185·29-s − 1.79·31-s + 1.31·37-s − 0.960·39-s − 0.468·41-s + 0.762·43-s + 1.16·47-s − 0.840·51-s + 0.824·53-s − 0.794·57-s − 0.260·59-s − 1.15·61-s − 0.855·67-s + 0.361·69-s − 0.712·71-s + 1.17·73-s + 1.12·79-s + 81-s − 0.987·83-s + 0.321·87-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$19600$$    =    $$2^{4} \cdot 5^{2} \cdot 7^{2}$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{19600} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 1 Selberg data = $(2,\ 19600,\ (\ :1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

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