Properties

 Label 1-105-105.47-r1-0-0 Degree $1$ Conductor $105$ Sign $0.813 - 0.581i$ Analytic cond. $11.2838$ Root an. cond. $11.2838$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + (−0.866 + 0.5i)2-s + (0.5 − 0.866i)4-s + i·8-s + (0.5 − 0.866i)11-s + i·13-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s + (−0.5 − 0.866i)19-s + i·22-s + (0.866 − 0.5i)23-s + (−0.5 − 0.866i)26-s + 29-s + (0.5 − 0.866i)31-s + (0.866 + 0.5i)32-s + 34-s + ⋯
 L(s)  = 1 + (−0.866 + 0.5i)2-s + (0.5 − 0.866i)4-s + i·8-s + (0.5 − 0.866i)11-s + i·13-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s + (−0.5 − 0.866i)19-s + i·22-s + (0.866 − 0.5i)23-s + (−0.5 − 0.866i)26-s + 29-s + (0.5 − 0.866i)31-s + (0.866 + 0.5i)32-s + 34-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.813 - 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.813 - 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$1$$ Conductor: $$105$$    =    $$3 \cdot 5 \cdot 7$$ Sign: $0.813 - 0.581i$ Analytic conductor: $$11.2838$$ Root analytic conductor: $$11.2838$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{105} (47, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 105,\ (1:\ ),\ 0.813 - 0.581i)$$

Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.9627814922 - 0.3086037039i$$ $$L(\frac12)$$ $$\approx$$ $$0.9627814922 - 0.3086037039i$$ $$L(1)$$ $$\approx$$ $$0.7598669203 + 0.008769799075i$$ $$L(1)$$ $$\approx$$ $$0.7598669203 + 0.008769799075i$$

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
5 $$1$$
7 $$1$$
good2 $$1 + (-0.866 + 0.5i)T$$
11 $$1 + (0.5 - 0.866i)T$$
13 $$1 + iT$$
17 $$1 + (-0.866 - 0.5i)T$$
19 $$1 + (-0.5 - 0.866i)T$$
23 $$1 + (0.866 - 0.5i)T$$
29 $$1 + T$$
31 $$1 + (0.5 - 0.866i)T$$
37 $$1 + (0.866 - 0.5i)T$$
41 $$1 + T$$
43 $$1 - iT$$
47 $$1 + (0.866 - 0.5i)T$$
53 $$1 + (-0.866 - 0.5i)T$$
59 $$1 + (0.5 - 0.866i)T$$
61 $$1 + (0.5 + 0.866i)T$$
67 $$1 + (-0.866 - 0.5i)T$$
71 $$1 - T$$
73 $$1 + (-0.866 - 0.5i)T$$
79 $$1 + (0.5 + 0.866i)T$$
83 $$1 - iT$$
89 $$1 + (0.5 + 0.866i)T$$
97 $$1 - iT$$
$$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$$