# Properties

 Degree 2 Conductor $5 \cdot 401$ Sign $-1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 1

# Origins

## Dirichlet series

 L(s)  = 1 − 2-s − 4-s + 5-s + 3·8-s − 3·9-s − 10-s + 4·11-s − 2·13-s − 16-s − 6·17-s + 3·18-s − 20-s − 4·22-s + 4·23-s + 25-s + 2·26-s − 2·29-s + 4·31-s − 5·32-s + 6·34-s + 3·36-s − 10·37-s + 3·40-s − 6·41-s + 4·43-s − 4·44-s − 3·45-s + ⋯
 L(s)  = 1 − 0.707·2-s − 1/2·4-s + 0.447·5-s + 1.06·8-s − 9-s − 0.316·10-s + 1.20·11-s − 0.554·13-s − 1/4·16-s − 1.45·17-s + 0.707·18-s − 0.223·20-s − 0.852·22-s + 0.834·23-s + 1/5·25-s + 0.392·26-s − 0.371·29-s + 0.718·31-s − 0.883·32-s + 1.02·34-s + 1/2·36-s − 1.64·37-s + 0.474·40-s − 0.937·41-s + 0.609·43-s − 0.603·44-s − 0.447·45-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$2005$$    =    $$5 \cdot 401$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{2005} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = 1 Selberg data = $(2,\ 2005,\ (\ :1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{5,\;401\}$, $F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{5,\;401\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad5 $$1 - T$$
401 $$1 - T$$
good2 $$1 + T + p T^{2}$$
3 $$1 + p T^{2}$$
7 $$1 + p T^{2}$$
11 $$1 - 4 T + p T^{2}$$
13 $$1 + 2 T + p T^{2}$$
17 $$1 + 6 T + p T^{2}$$
19 $$1 + p T^{2}$$
23 $$1 - 4 T + p T^{2}$$
29 $$1 + 2 T + p T^{2}$$
31 $$1 - 4 T + p T^{2}$$
37 $$1 + 10 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 - 6 T + p T^{2}$$
59 $$1 - 8 T + p T^{2}$$
61 $$1 + 10 T + p T^{2}$$
67 $$1 + p T^{2}$$
71 $$1 - 4 T + p T^{2}$$
73 $$1 + 6 T + p T^{2}$$
79 $$1 + 12 T + p T^{2}$$
83 $$1 + 12 T + p T^{2}$$
89 $$1 - 10 T + p T^{2}$$
97 $$1 + 6 T + p T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}