# Properties

 Label 2-23-23.22-c0-0-0 Degree $2$ Conductor $23$ Sign $1$ Analytic cond. $0.0114784$ Root an. cond. $0.107137$ Motivic weight $0$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

This L-function has the smallest analytic conductor among primitive algebraic degree 2 L-functions.

## Dirichlet series

 L(s)  = 1 − 2-s − 3-s + 6-s + 8-s − 13-s − 16-s + 23-s − 24-s + 25-s + 26-s + 27-s − 29-s − 31-s + 39-s − 41-s − 46-s − 47-s + 48-s + 49-s − 50-s − 54-s + 58-s + 2·59-s + 62-s + 64-s − 69-s − 71-s + ⋯
 L(s)  = 1 − 2-s − 3-s + 6-s + 8-s − 13-s − 16-s + 23-s − 24-s + 25-s + 26-s + 27-s − 29-s − 31-s + 39-s − 41-s − 46-s − 47-s + 48-s + 49-s − 50-s − 54-s + 58-s + 2·59-s + 62-s + 64-s − 69-s − 71-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$23$$ Sign: $1$ Analytic conductor: $$0.0114784$$ Root analytic conductor: $$0.107137$$ Motivic weight: $$0$$ Rational: yes Arithmetic: yes Character: $\chi_{23} (22, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 23,\ (\ :0),\ 1)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.1740363269$$ $$L(\frac12)$$ $$\approx$$ $$0.1740363269$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad23 $$1 - T$$
good2 $$1 + T + T^{2}$$
3 $$1 + T + T^{2}$$
5 $$( 1 - T )( 1 + T )$$
7 $$( 1 - T )( 1 + T )$$
11 $$( 1 - T )( 1 + T )$$
13 $$1 + T + T^{2}$$
17 $$( 1 - T )( 1 + T )$$
19 $$( 1 - T )( 1 + T )$$
29 $$1 + T + T^{2}$$
31 $$1 + T + T^{2}$$
37 $$( 1 - T )( 1 + T )$$
41 $$1 + T + T^{2}$$
43 $$( 1 - T )( 1 + T )$$
47 $$1 + T + T^{2}$$
53 $$( 1 - T )( 1 + T )$$
59 $$( 1 - T )^{2}$$
61 $$( 1 - T )( 1 + T )$$
67 $$( 1 - T )( 1 + T )$$
71 $$1 + T + T^{2}$$
73 $$1 + T + T^{2}$$
79 $$( 1 - T )( 1 + T )$$
83 $$( 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}$$