# Properties

 Label 1-10e2-100.47-r0-0-0 Degree $1$ Conductor $100$ Sign $0.998 - 0.0627i$ Analytic cond. $0.464398$ Root an. cond. $0.464398$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.951 + 0.309i)3-s − i·7-s + (0.809 − 0.587i)9-s + (0.809 + 0.587i)11-s + (0.587 + 0.809i)13-s + (0.951 + 0.309i)17-s + (0.309 − 0.951i)19-s + (0.309 + 0.951i)21-s + (0.587 − 0.809i)23-s + (−0.587 + 0.809i)27-s + (−0.309 − 0.951i)29-s + (−0.309 + 0.951i)31-s + (−0.951 − 0.309i)33-s + (−0.587 − 0.809i)37-s + (−0.809 − 0.587i)39-s + ⋯
 L(s)  = 1 + (−0.951 + 0.309i)3-s − i·7-s + (0.809 − 0.587i)9-s + (0.809 + 0.587i)11-s + (0.587 + 0.809i)13-s + (0.951 + 0.309i)17-s + (0.309 − 0.951i)19-s + (0.309 + 0.951i)21-s + (0.587 − 0.809i)23-s + (−0.587 + 0.809i)27-s + (−0.309 − 0.951i)29-s + (−0.309 + 0.951i)31-s + (−0.951 − 0.309i)33-s + (−0.587 − 0.809i)37-s + (−0.809 − 0.587i)39-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0627i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0627i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$100$$    =    $$2^{2} \cdot 5^{2}$$ Sign: $0.998 - 0.0627i$ Analytic conductor: $$0.464398$$ Root analytic conductor: $$0.464398$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{100} (47, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 100,\ (0:\ ),\ 0.998 - 0.0627i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.8189813763 + 0.02573752661i$$ $$L(\frac12)$$ $$\approx$$ $$0.8189813763 + 0.02573752661i$$ $$L(1)$$ $$\approx$$ $$0.8632529409 + 0.002418697518i$$ $$L(1)$$ $$\approx$$ $$0.8632529409 + 0.002418697518i$$

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$1$$
good3 $$1 + (-0.951 + 0.309i)T$$
7 $$1 - iT$$
11 $$1 + (0.809 + 0.587i)T$$
13 $$1 + (0.587 + 0.809i)T$$
17 $$1 + (0.951 + 0.309i)T$$
19 $$1 + (0.309 - 0.951i)T$$
23 $$1 + (0.587 - 0.809i)T$$
29 $$1 + (-0.309 - 0.951i)T$$
31 $$1 + (-0.309 + 0.951i)T$$
37 $$1 + (-0.587 - 0.809i)T$$
41 $$1 + (-0.809 + 0.587i)T$$
43 $$1 - iT$$
47 $$1 + (0.951 - 0.309i)T$$
53 $$1 + (0.951 - 0.309i)T$$
59 $$1 + (-0.809 + 0.587i)T$$
61 $$1 + (-0.809 - 0.587i)T$$
67 $$1 + (-0.951 - 0.309i)T$$
71 $$1 + (-0.309 - 0.951i)T$$
73 $$1 + (-0.587 + 0.809i)T$$
79 $$1 + (0.309 + 0.951i)T$$
83 $$1 + (0.951 + 0.309i)T$$
89 $$1 + (0.809 + 0.587i)T$$
97 $$1 + (-0.951 + 0.309i)T$$
$$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$$