# Properties

 Degree $2$ Conductor $4080$ Sign $-1$ Motivic weight $1$ Primitive yes Self-dual yes Analytic rank $1$

# Origins

## Dirichlet series

 L(s)  = 1 + 3-s + 5-s + 9-s − 4·11-s − 2·13-s + 15-s + 17-s − 4·19-s + 25-s + 27-s − 2·29-s − 8·31-s − 4·33-s + 6·37-s − 2·39-s − 6·41-s + 4·43-s + 45-s − 7·49-s + 51-s − 10·53-s − 4·55-s − 4·57-s + 4·59-s − 2·61-s − 2·65-s − 4·67-s + ⋯
 L(s)  = 1 + 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.20·11-s − 0.554·13-s + 0.258·15-s + 0.242·17-s − 0.917·19-s + 1/5·25-s + 0.192·27-s − 0.371·29-s − 1.43·31-s − 0.696·33-s + 0.986·37-s − 0.320·39-s − 0.937·41-s + 0.609·43-s + 0.149·45-s − 49-s + 0.140·51-s − 1.37·53-s − 0.539·55-s − 0.529·57-s + 0.520·59-s − 0.256·61-s − 0.248·65-s − 0.488·67-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$4080$$    =    $$2^{4} \cdot 3 \cdot 5 \cdot 17$$ Sign: $-1$ Motivic weight: $$1$$ Character: $\chi_{4080} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 4080,\ (\ :1/2),\ -1)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1 - T$$
5 $$1 - T$$
17 $$1 - T$$
good7 $$1 + p T^{2}$$
11 $$1 + 4 T + p T^{2}$$
13 $$1 + 2 T + p T^{2}$$
19 $$1 + 4 T + p T^{2}$$
23 $$1 + 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 - 4 T + p T^{2}$$
47 $$1 + p T^{2}$$
53 $$1 + 10 T + p T^{2}$$
59 $$1 - 4 T + p T^{2}$$
61 $$1 + 2 T + p T^{2}$$
67 $$1 + 4 T + p T^{2}$$
71 $$1 + p T^{2}$$
73 $$1 + 6 T + p T^{2}$$
79 $$1 + 8 T + p T^{2}$$
83 $$1 - 12 T + p T^{2}$$
89 $$1 + 6 T + p T^{2}$$
97 $$1 + 14 T + p T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$