# Properties

 Label 2-4080-1.1-c1-0-18 Degree $2$ Conductor $4080$ Sign $1$ Analytic cond. $32.5789$ Root an. cond. $5.70779$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 + 3-s − 5-s − 2·7-s + 9-s + 4·11-s + 4·13-s − 15-s + 17-s + 4·19-s − 2·21-s − 8·23-s + 25-s + 27-s + 2·29-s − 4·31-s + 4·33-s + 2·35-s + 6·37-s + 4·39-s + 8·41-s − 6·43-s − 45-s − 8·47-s − 3·49-s + 51-s − 2·53-s − 4·55-s + ⋯
 L(s)  = 1 + 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s + 1.20·11-s + 1.10·13-s − 0.258·15-s + 0.242·17-s + 0.917·19-s − 0.436·21-s − 1.66·23-s + 1/5·25-s + 0.192·27-s + 0.371·29-s − 0.718·31-s + 0.696·33-s + 0.338·35-s + 0.986·37-s + 0.640·39-s + 1.24·41-s − 0.914·43-s − 0.149·45-s − 1.16·47-s − 3/7·49-s + 0.140·51-s − 0.274·53-s − 0.539·55-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: $1$ Analytic conductor: $$32.5789$$ Root analytic conductor: $$5.70779$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 4080,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.279579886$$ $$L(\frac12)$$ $$\approx$$ $$2.279579886$$ $$L(\frac{3}{2})$$ 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 + 2 T + p T^{2}$$
11 $$1 - 4 T + p T^{2}$$
13 $$1 - 4 T + p T^{2}$$
19 $$1 - 4 T + p T^{2}$$
23 $$1 + 8 T + p T^{2}$$
29 $$1 - 2 T + p T^{2}$$
31 $$1 + 4 T + p T^{2}$$
37 $$1 - 6 T + p T^{2}$$
41 $$1 - 8 T + p T^{2}$$
43 $$1 + 6 T + p T^{2}$$
47 $$1 + 8 T + p T^{2}$$
53 $$1 + 2 T + p T^{2}$$
59 $$1 - 6 T + p T^{2}$$
61 $$1 - 14 T + p T^{2}$$
67 $$1 - 2 T + p T^{2}$$
71 $$1 + 2 T + p T^{2}$$
73 $$1 - 4 T + p T^{2}$$
79 $$1 + p T^{2}$$
83 $$1 - 16 T + p T^{2}$$
89 $$1 + 2 T + p T^{2}$$
97 $$1 + p T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$