# Properties

 Degree 2 Conductor $2^{3} \cdot 503$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 2

# Origins

## Dirichlet series

 L(s)  = 1 − 3-s − 5·7-s − 2·9-s − 5·11-s + 13-s − 4·17-s − 4·19-s + 5·21-s − 9·23-s − 5·25-s + 5·27-s + 2·29-s + 2·31-s + 5·33-s − 6·37-s − 39-s + 6·41-s − 5·43-s + 47-s + 18·49-s + 4·51-s − 6·53-s + 4·57-s − 12·59-s − 3·61-s + 10·63-s − 5·67-s + ⋯
 L(s)  = 1 − 0.577·3-s − 1.88·7-s − 2/3·9-s − 1.50·11-s + 0.277·13-s − 0.970·17-s − 0.917·19-s + 1.09·21-s − 1.87·23-s − 25-s + 0.962·27-s + 0.371·29-s + 0.359·31-s + 0.870·33-s − 0.986·37-s − 0.160·39-s + 0.937·41-s − 0.762·43-s + 0.145·47-s + 18/7·49-s + 0.560·51-s − 0.824·53-s + 0.529·57-s − 1.56·59-s − 0.384·61-s + 1.25·63-s − 0.610·67-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$4024$$    =    $$2^{3} \cdot 503$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{4024} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = 2 Selberg data = $(2,\ 4024,\ (\ :1/2),\ 1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;503\}$, $F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{2,\;503\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 $$1$$
503 $$1 + T$$
good3 $$1 + T + p T^{2}$$
5 $$1 + p T^{2}$$
7 $$1 + 5 T + p T^{2}$$
11 $$1 + 5 T + p T^{2}$$
13 $$1 - T + p T^{2}$$
17 $$1 + 4 T + p T^{2}$$
19 $$1 + 4 T + p T^{2}$$
23 $$1 + 9 T + p T^{2}$$
29 $$1 - 2 T + p T^{2}$$
31 $$1 - 2 T + p T^{2}$$
37 $$1 + 6 T + p T^{2}$$
41 $$1 - 6 T + p T^{2}$$
43 $$1 + 5 T + p T^{2}$$
47 $$1 - T + p T^{2}$$
53 $$1 + 6 T + p T^{2}$$
59 $$1 + 12 T + p T^{2}$$
61 $$1 + 3 T + p T^{2}$$
67 $$1 + 5 T + p T^{2}$$
71 $$1 - 6 T + p T^{2}$$
73 $$1 + 2 T + p T^{2}$$
79 $$1 + 8 T + p T^{2}$$
83 $$1 - 7 T + p T^{2}$$
89 $$1 - 6 T + p T^{2}$$
97 $$1 - 10 T + p T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}