# Properties

 Label 2-87-87.86-c6-0-37 Degree $2$ Conductor $87$ Sign $1$ Analytic cond. $20.0147$ Root an. cond. $4.47377$ Motivic weight $6$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 + 13·2-s − 27·3-s + 105·4-s − 351·6-s + 338·7-s + 533·8-s + 729·9-s − 806·11-s − 2.83e3·12-s + 3.00e3·13-s + 4.39e3·14-s + 209·16-s + 9.77e3·17-s + 9.47e3·18-s − 9.12e3·21-s − 1.04e4·22-s − 1.43e4·24-s + 1.56e4·25-s + 3.90e4·26-s − 1.96e4·27-s + 3.54e4·28-s + 2.43e4·29-s − 3.13e4·32-s + 2.17e4·33-s + 1.27e5·34-s + 7.65e4·36-s − 8.10e4·39-s + ⋯
 L(s)  = 1 + 13/8·2-s − 3-s + 1.64·4-s − 1.62·6-s + 0.985·7-s + 1.04·8-s + 9-s − 0.605·11-s − 1.64·12-s + 1.36·13-s + 1.60·14-s + 0.0510·16-s + 1.99·17-s + 13/8·18-s − 0.985·21-s − 0.984·22-s − 1.04·24-s + 25-s + 2.22·26-s − 27-s + 1.61·28-s + 29-s − 0.958·32-s + 0.605·33-s + 3.23·34-s + 1.64·36-s − 1.36·39-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$87$$    =    $$3 \cdot 29$$ Sign: $1$ Analytic conductor: $$20.0147$$ Root analytic conductor: $$4.47377$$ Motivic weight: $$6$$ Rational: yes Arithmetic: yes Character: $\chi_{87} (86, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 87,\ (\ :3),\ 1)$$

## Particular Values

 $$L(\frac{7}{2})$$ $$\approx$$ $$4.072483265$$ $$L(\frac12)$$ $$\approx$$ $$4.072483265$$ $$L(4)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + p^{3} T$$
29 $$1 - p^{3} T$$
good2 $$1 - 13 T + p^{6} T^{2}$$
5 $$( 1 - p^{3} T )( 1 + p^{3} T )$$
7 $$1 - 338 T + p^{6} T^{2}$$
11 $$1 + 806 T + p^{6} T^{2}$$
13 $$1 - 3002 T + p^{6} T^{2}$$
17 $$1 - 9778 T + p^{6} T^{2}$$
19 $$( 1 - p^{3} T )( 1 + p^{3} T )$$
23 $$( 1 - p^{3} T )( 1 + p^{3} T )$$
31 $$( 1 - p^{3} T )( 1 + p^{3} T )$$
37 $$( 1 - p^{3} T )( 1 + p^{3} T )$$
41 $$1 + 132158 T + p^{6} T^{2}$$
43 $$( 1 - p^{3} T )( 1 + p^{3} T )$$
47 $$1 + 151502 T + p^{6} T^{2}$$
53 $$( 1 - p^{3} T )( 1 + p^{3} T )$$
59 $$( 1 - p^{3} T )( 1 + p^{3} T )$$
61 $$( 1 - p^{3} T )( 1 + p^{3} T )$$
67 $$1 - 267098 T + p^{6} T^{2}$$
71 $$( 1 - p^{3} T )( 1 + p^{3} T )$$
73 $$( 1 - p^{3} T )( 1 + p^{3} T )$$
79 $$( 1 - p^{3} T )( 1 + p^{3} T )$$
83 $$( 1 - p^{3} T )( 1 + p^{3} T )$$
89 $$1 - 71266 T + p^{6} T^{2}$$
97 $$( 1 - p^{3} T )( 1 + p^{3} T )$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$