# Properties

 Degree $2$ Conductor $59$ Sign $1$ Motivic weight $0$ Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 − 3-s + 4-s − 5-s − 7-s − 12-s + 15-s + 16-s + 2·17-s − 19-s − 20-s + 21-s + 27-s − 28-s − 29-s + 35-s − 41-s − 48-s − 2·51-s − 53-s + 57-s + 59-s + 60-s + 64-s + 2·68-s + 2·71-s − 76-s − 79-s + ⋯
 L(s)  = 1 − 3-s + 4-s − 5-s − 7-s − 12-s + 15-s + 16-s + 2·17-s − 19-s − 20-s + 21-s + 27-s − 28-s − 29-s + 35-s − 41-s − 48-s − 2·51-s − 53-s + 57-s + 59-s + 60-s + 64-s + 2·68-s + 2·71-s − 76-s − 79-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$59$$ Sign: $1$ Motivic weight: $$0$$ Character: $\chi_{59} (58, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 59,\ (\ :0),\ 1)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.3770726116$$ $$L(\frac12)$$ $$\approx$$ $$0.3770726116$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad59 $$1 - T$$
good2 $$( 1 - T )( 1 + T )$$
3 $$1 + T + T^{2}$$
5 $$1 + T + T^{2}$$
7 $$1 + T + T^{2}$$
11 $$( 1 - T )( 1 + T )$$
13 $$( 1 - T )( 1 + T )$$
17 $$( 1 - T )^{2}$$
19 $$1 + T + T^{2}$$
23 $$( 1 - T )( 1 + T )$$
29 $$1 + T + T^{2}$$
31 $$( 1 - T )( 1 + T )$$
37 $$( 1 - T )( 1 + T )$$
41 $$1 + T + T^{2}$$
43 $$( 1 - T )( 1 + T )$$
47 $$( 1 - T )( 1 + T )$$
53 $$1 + T + T^{2}$$
61 $$( 1 - T )( 1 + T )$$
67 $$( 1 - T )( 1 + T )$$
71 $$( 1 - T )^{2}$$
73 $$( 1 - T )( 1 + T )$$
79 $$1 + T + T^{2}$$
83 $$( 1 - T )( 1 + T )$$
89 $$( 1 - T )( 1 + T )$$
97 $$( 1 - T )( 1 + T )$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$