Properties

 Degree 2 Conductor $2^{6} \cdot 5^{2}$ Sign $-1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 1

Related objects

Dirichlet series

 L(s)  = 1 + 4·7-s − 3·9-s − 4·11-s − 2·13-s − 2·17-s − 4·19-s − 4·23-s + 2·29-s − 8·31-s + 6·37-s − 6·41-s − 8·43-s − 4·47-s + 9·49-s + 6·53-s + 4·59-s + 2·61-s − 12·63-s + 8·67-s + 6·73-s − 16·77-s + 9·81-s − 16·83-s − 6·89-s − 8·91-s + 14·97-s + 12·99-s + ⋯
 L(s)  = 1 + 1.51·7-s − 9-s − 1.20·11-s − 0.554·13-s − 0.485·17-s − 0.917·19-s − 0.834·23-s + 0.371·29-s − 1.43·31-s + 0.986·37-s − 0.937·41-s − 1.21·43-s − 0.583·47-s + 9/7·49-s + 0.824·53-s + 0.520·59-s + 0.256·61-s − 1.51·63-s + 0.977·67-s + 0.702·73-s − 1.82·77-s + 81-s − 1.75·83-s − 0.635·89-s − 0.838·91-s + 1.42·97-s + 1.20·99-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

 $$d$$ = $$2$$ $$N$$ = $$1600$$    =    $$2^{6} \cdot 5^{2}$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{1600} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 1 Selberg data = $(2,\ 1600,\ (\ :1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;5\}$,$F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{2,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
5 $$1$$
good3 $$1 + p T^{2}$$
7 $$1 - 4 T + p T^{2}$$
11 $$1 + 4 T + p T^{2}$$
13 $$1 + 2 T + p T^{2}$$
17 $$1 + 2 T + p T^{2}$$
19 $$1 + 4 T + p T^{2}$$
23 $$1 + 4 T + p T^{2}$$
29 $$1 - 2 T + p T^{2}$$
31 $$1 + 8 T + p T^{2}$$
37 $$1 - 6 T + p T^{2}$$
41 $$1 + 6 T + p T^{2}$$
43 $$1 + 8 T + p T^{2}$$
47 $$1 + 4 T + p T^{2}$$
53 $$1 - 6 T + p T^{2}$$
59 $$1 - 4 T + p T^{2}$$
61 $$1 - 2 T + p T^{2}$$
67 $$1 - 8 T + p T^{2}$$
71 $$1 + p T^{2}$$
73 $$1 - 6 T + p T^{2}$$
79 $$1 + p T^{2}$$
83 $$1 + 16 T + p T^{2}$$
89 $$1 + 6 T + p T^{2}$$
97 $$1 - 14 T + p T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}