# Properties

 Degree $2$ Conductor $113$ Sign $1$ Motivic weight $1$ Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2-s + 2·3-s − 4-s + 2·5-s − 2·6-s + 3·8-s + 9-s − 2·10-s − 2·12-s + 2·13-s + 4·15-s − 16-s − 6·17-s − 18-s + 6·19-s − 2·20-s − 6·23-s + 6·24-s − 25-s − 2·26-s − 4·27-s − 6·29-s − 4·30-s − 4·31-s − 5·32-s + 6·34-s − 36-s + ⋯
 L(s)  = 1 − 0.707·2-s + 1.15·3-s − 1/2·4-s + 0.894·5-s − 0.816·6-s + 1.06·8-s + 1/3·9-s − 0.632·10-s − 0.577·12-s + 0.554·13-s + 1.03·15-s − 1/4·16-s − 1.45·17-s − 0.235·18-s + 1.37·19-s − 0.447·20-s − 1.25·23-s + 1.22·24-s − 1/5·25-s − 0.392·26-s − 0.769·27-s − 1.11·29-s − 0.730·30-s − 0.718·31-s − 0.883·32-s + 1.02·34-s − 1/6·36-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$113$$ Sign: $1$ Motivic weight: $$1$$ Character: $\chi_{113} (1, \cdot )$ Sato-Tate group: $\mathrm{SU}(2)$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 113,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.009184946$$ $$L(\frac12)$$ $$\approx$$ $$1.009184946$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad113 $$1 - T$$
good2 $$1 + T + p T^{2}$$
3 $$1 - 2 T + p T^{2}$$
5 $$1 - 2 T + p T^{2}$$
7 $$1 + p T^{2}$$
11 $$1 + p T^{2}$$
13 $$1 - 2 T + p T^{2}$$
17 $$1 + 6 T + p T^{2}$$
19 $$1 - 6 T + p T^{2}$$
23 $$1 + 6 T + p T^{2}$$
29 $$1 + 6 T + p T^{2}$$
31 $$1 + 4 T + p T^{2}$$
37 $$1 - 2 T + p T^{2}$$
41 $$1 + 2 T + p T^{2}$$
43 $$1 - 6 T + p T^{2}$$
47 $$1 - 6 T + p T^{2}$$
53 $$1 - 10 T + p T^{2}$$
59 $$1 - 6 T + p T^{2}$$
61 $$1 - 6 T + p T^{2}$$
67 $$1 - 2 T + p T^{2}$$
71 $$1 + 6 T + p T^{2}$$
73 $$1 - 2 T + p T^{2}$$
79 $$1 - 10 T + p T^{2}$$
83 $$1 + 4 T + p T^{2}$$
89 $$1 + 14 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}$$