# Properties

 Label 1-1375-1375.294-r1-0-0 Degree $1$ Conductor $1375$ Sign $-0.498 + 0.866i$ Analytic cond. $147.764$ Root an. cond. $147.764$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.876 + 0.481i)2-s + (0.637 − 0.770i)3-s + (0.535 + 0.844i)4-s + (0.929 − 0.368i)6-s + (−0.809 + 0.587i)7-s + (0.0627 + 0.998i)8-s + (−0.187 − 0.982i)9-s + (0.992 + 0.125i)12-s + (−0.187 − 0.982i)13-s + (−0.992 + 0.125i)14-s + (−0.425 + 0.904i)16-s + (−0.637 − 0.770i)17-s + (0.309 − 0.951i)18-s + (0.929 − 0.368i)19-s + (−0.0627 + 0.998i)21-s + ⋯
 L(s)  = 1 + (0.876 + 0.481i)2-s + (0.637 − 0.770i)3-s + (0.535 + 0.844i)4-s + (0.929 − 0.368i)6-s + (−0.809 + 0.587i)7-s + (0.0627 + 0.998i)8-s + (−0.187 − 0.982i)9-s + (0.992 + 0.125i)12-s + (−0.187 − 0.982i)13-s + (−0.992 + 0.125i)14-s + (−0.425 + 0.904i)16-s + (−0.637 − 0.770i)17-s + (0.309 − 0.951i)18-s + (0.929 − 0.368i)19-s + (−0.0627 + 0.998i)21-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.498 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.498 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$1375$$    =    $$5^{3} \cdot 11$$ Sign: $-0.498 + 0.866i$ Analytic conductor: $$147.764$$ Root analytic conductor: $$147.764$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{1375} (294, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 1375,\ (1:\ ),\ -0.498 + 0.866i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.349420942 + 2.332749222i$$ $$L(\frac12)$$ $$\approx$$ $$1.349420942 + 2.332749222i$$ $$L(1)$$ $$\approx$$ $$1.709411747 + 0.4036549796i$$ $$L(1)$$ $$\approx$$ $$1.709411747 + 0.4036549796i$$

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1$$
11 $$1$$
good2 $$1 + (0.876 + 0.481i)T$$
3 $$1 + (0.637 - 0.770i)T$$
7 $$1 + (-0.809 + 0.587i)T$$
13 $$1 + (-0.187 - 0.982i)T$$
17 $$1 + (-0.637 - 0.770i)T$$
19 $$1 + (0.929 - 0.368i)T$$
23 $$1 + (0.425 + 0.904i)T$$
29 $$1 + (-0.968 + 0.248i)T$$
31 $$1 + (0.0627 + 0.998i)T$$
37 $$1 + (0.187 + 0.982i)T$$
41 $$1 + (0.187 + 0.982i)T$$
43 $$1 + (-0.809 - 0.587i)T$$
47 $$1 + (0.929 + 0.368i)T$$
53 $$1 + (0.929 + 0.368i)T$$
59 $$1 + (0.728 + 0.684i)T$$
61 $$1 + (-0.728 + 0.684i)T$$
67 $$1 + (0.637 + 0.770i)T$$
71 $$1 + (-0.637 + 0.770i)T$$
73 $$1 + (0.876 + 0.481i)T$$
79 $$1 + (-0.0627 + 0.998i)T$$
83 $$1 + (0.0627 + 0.998i)T$$
89 $$1 + (-0.992 + 0.125i)T$$
97 $$1 + (-0.535 - 0.844i)T$$
$$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$$