# Properties

 Degree 2 Conductor 4013 Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Origins

## Dirichlet series

 L(s)  = 1 + 2·2-s − 2·3-s + 2·4-s + 4·5-s − 4·6-s + 7-s + 9-s + 8·10-s + 3·11-s − 4·12-s + 2·13-s + 2·14-s − 8·15-s − 4·16-s − 2·17-s + 2·18-s + 19-s + 8·20-s − 2·21-s + 6·22-s + 11·25-s + 4·26-s + 4·27-s + 2·28-s + 8·29-s − 16·30-s − 5·31-s + ⋯
 L(s)  = 1 + 1.41·2-s − 1.15·3-s + 4-s + 1.78·5-s − 1.63·6-s + 0.377·7-s + 1/3·9-s + 2.52·10-s + 0.904·11-s − 1.15·12-s + 0.554·13-s + 0.534·14-s − 2.06·15-s − 16-s − 0.485·17-s + 0.471·18-s + 0.229·19-s + 1.78·20-s − 0.436·21-s + 1.27·22-s + 11/5·25-s + 0.784·26-s + 0.769·27-s + 0.377·28-s + 1.48·29-s − 2.92·30-s − 0.898·31-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 4013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 4013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$4013$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{4013} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 4013,\ (\ :1/2),\ 1)$ $L(1)$ $\approx$ $4.224266485$ $L(\frac12)$ $\approx$ $4.224266485$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 4013$, $F_p(T) = 1 - a_p T + p T^2 .$If $p = 4013$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad4013 $$1+O(T)$$
good2 $$1 - p T + p T^{2}$$
3 $$1 + 2 T + p T^{2}$$
5 $$1 - 4 T + p T^{2}$$
7 $$1 - T + p T^{2}$$
11 $$1 - 3 T + p T^{2}$$
13 $$1 - 2 T + p T^{2}$$
17 $$1 + 2 T + p T^{2}$$
19 $$1 - T + p T^{2}$$
23 $$1 + p T^{2}$$
29 $$1 - 8 T + p T^{2}$$
31 $$1 + 5 T + p T^{2}$$
37 $$1 + 4 T + p T^{2}$$
41 $$1 - 5 T + p T^{2}$$
43 $$1 - 5 T + p T^{2}$$
47 $$1 - 12 T + p T^{2}$$
53 $$1 - T + p T^{2}$$
59 $$1 - 7 T + p T^{2}$$
61 $$1 + 7 T + p T^{2}$$
67 $$1 + 7 T + p T^{2}$$
71 $$1 - 4 T + p T^{2}$$
73 $$1 + 13 T + p T^{2}$$
79 $$1 - 4 T + p T^{2}$$
83 $$1 - 12 T + p T^{2}$$
89 $$1 + 10 T + p T^{2}$$
97 $$1 + 17 T + p T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}