# Properties

 Label 2-87120-1.1-c1-0-118 Degree $2$ Conductor $87120$ Sign $-1$ Analytic cond. $695.656$ Root an. cond. $26.3753$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

# Origins

## Dirichlet series

 L(s)  = 1 + 5-s − 4·7-s + 2·13-s − 5·17-s + 8·19-s + 5·23-s + 25-s + 6·29-s + 9·31-s − 4·35-s − 4·37-s + 6·41-s + 6·43-s − 13·47-s + 9·49-s − 9·53-s − 10·59-s − 11·61-s + 2·65-s − 12·67-s + 8·71-s + 4·73-s + 5·79-s − 8·83-s − 5·85-s + 8·89-s − 8·91-s + ⋯
 L(s)  = 1 + 0.447·5-s − 1.51·7-s + 0.554·13-s − 1.21·17-s + 1.83·19-s + 1.04·23-s + 1/5·25-s + 1.11·29-s + 1.61·31-s − 0.676·35-s − 0.657·37-s + 0.937·41-s + 0.914·43-s − 1.89·47-s + 9/7·49-s − 1.23·53-s − 1.30·59-s − 1.40·61-s + 0.248·65-s − 1.46·67-s + 0.949·71-s + 0.468·73-s + 0.562·79-s − 0.878·83-s − 0.542·85-s + 0.847·89-s − 0.838·91-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$87120$$    =    $$2^{4} \cdot 3^{2} \cdot 5 \cdot 11^{2}$$ Sign: $-1$ Analytic conductor: $$695.656$$ Root analytic conductor: $$26.3753$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{87120} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 87120,\ (\ :1/2),\ -1)$$

## 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$$
5 $$1 - T$$
11 $$1$$
good7 $$1 + 4 T + p T^{2}$$
13 $$1 - 2 T + p T^{2}$$
17 $$1 + 5 T + p T^{2}$$
19 $$1 - 8 T + p T^{2}$$
23 $$1 - 5 T + p T^{2}$$
29 $$1 - 6 T + p T^{2}$$
31 $$1 - 9 T + p T^{2}$$
37 $$1 + 4 T + p T^{2}$$
41 $$1 - 6 T + p T^{2}$$
43 $$1 - 6 T + p T^{2}$$
47 $$1 + 13 T + p T^{2}$$
53 $$1 + 9 T + p T^{2}$$
59 $$1 + 10 T + p T^{2}$$
61 $$1 + 11 T + p T^{2}$$
67 $$1 + 12 T + p T^{2}$$
71 $$1 - 8 T + p T^{2}$$
73 $$1 - 4 T + p T^{2}$$
79 $$1 - 5 T + p T^{2}$$
83 $$1 + 8 T + p T^{2}$$
89 $$1 - 8 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}$$