# Properties

 Label 2-52-52.3-c0-0-0 Degree $2$ Conductor $52$ Sign $0.711 + 0.702i$ Analytic cond. $0.0259513$ Root an. cond. $0.161094$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

This L-function's analytic conductor is the smallest among non-self-dual primitive algebraic degree 2 L-functions

## Dirichlet series

 L(s)  = 1 + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s − 5-s + 0.999·8-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)10-s + (−0.5 − 0.866i)13-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + 0.999·18-s + (0.499 − 0.866i)20-s + (−0.499 + 0.866i)26-s + (0.5 + 0.866i)29-s + (−0.499 + 0.866i)32-s − 0.999·34-s + ⋯
 L(s)  = 1 + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s − 5-s + 0.999·8-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)10-s + (−0.5 − 0.866i)13-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + 0.999·18-s + (0.499 − 0.866i)20-s + (−0.499 + 0.866i)26-s + (0.5 + 0.866i)29-s + (−0.499 + 0.866i)32-s − 0.999·34-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$52$$    =    $$2^{2} \cdot 13$$ Sign: $0.711 + 0.702i$ Analytic conductor: $$0.0259513$$ Root analytic conductor: $$0.161094$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{52} (3, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 52,\ (\ :0),\ 0.711 + 0.702i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.3248332447$$ $$L(\frac12)$$ $$\approx$$ $$0.3248332447$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (0.5 + 0.866i)T$$
13 $$1 + (0.5 + 0.866i)T$$
good3 $$1 + (0.5 - 0.866i)T^{2}$$
5 $$1 + T + T^{2}$$
7 $$1 + (0.5 + 0.866i)T^{2}$$
11 $$1 + (0.5 - 0.866i)T^{2}$$
17 $$1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}$$
19 $$1 + (0.5 + 0.866i)T^{2}$$
23 $$1 + (0.5 - 0.866i)T^{2}$$
29 $$1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}$$
31 $$1 - T^{2}$$
37 $$1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}$$
41 $$1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}$$
43 $$1 + (0.5 + 0.866i)T^{2}$$
47 $$1 - T^{2}$$
53 $$1 + T + T^{2}$$
59 $$1 + (0.5 + 0.866i)T^{2}$$
61 $$1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}$$
67 $$1 + (0.5 - 0.866i)T^{2}$$
71 $$1 + (0.5 + 0.866i)T^{2}$$
73 $$1 + T + T^{2}$$
79 $$1 - T^{2}$$
83 $$1 - T^{2}$$
89 $$1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2}$$
97 $$1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$