# Properties

 Label 1-1441-1441.1080-r0-0-0 Degree $1$ Conductor $1441$ Sign $0.881 - 0.471i$ Analytic cond. $6.69197$ Root an. cond. $6.69197$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.644 + 0.764i)2-s + (−0.906 − 0.421i)3-s + (−0.168 + 0.985i)4-s + (0.485 + 0.873i)5-s + (−0.262 − 0.964i)6-s + (−0.779 + 0.626i)7-s + (−0.861 + 0.506i)8-s + (0.644 + 0.764i)9-s + (−0.354 + 0.935i)10-s + (0.568 − 0.822i)12-s + (0.262 − 0.964i)13-s + (−0.981 − 0.192i)14-s + (−0.0724 − 0.997i)15-s + (−0.943 − 0.331i)16-s + (−0.943 − 0.331i)17-s + (−0.168 + 0.985i)18-s + ⋯
 L(s)  = 1 + (0.644 + 0.764i)2-s + (−0.906 − 0.421i)3-s + (−0.168 + 0.985i)4-s + (0.485 + 0.873i)5-s + (−0.262 − 0.964i)6-s + (−0.779 + 0.626i)7-s + (−0.861 + 0.506i)8-s + (0.644 + 0.764i)9-s + (−0.354 + 0.935i)10-s + (0.568 − 0.822i)12-s + (0.262 − 0.964i)13-s + (−0.981 − 0.192i)14-s + (−0.0724 − 0.997i)15-s + (−0.943 − 0.331i)16-s + (−0.943 − 0.331i)17-s + (−0.168 + 0.985i)18-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$1441$$    =    $$11 \cdot 131$$ Sign: $0.881 - 0.471i$ Analytic conductor: $$6.69197$$ Root analytic conductor: $$6.69197$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{1441} (1080, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 1441,\ (0:\ ),\ 0.881 - 0.471i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.6911881672 - 0.1732452820i$$ $$L(\frac12)$$ $$\approx$$ $$0.6911881672 - 0.1732452820i$$ $$L(1)$$ $$\approx$$ $$0.8171929423 + 0.3599850092i$$ $$L(1)$$ $$\approx$$ $$0.8171929423 + 0.3599850092i$$

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad11 $$1$$
131 $$1$$
good2 $$1 + (0.644 + 0.764i)T$$
3 $$1 + (-0.906 - 0.421i)T$$
5 $$1 + (0.485 + 0.873i)T$$
7 $$1 + (-0.779 + 0.626i)T$$
13 $$1 + (0.262 - 0.964i)T$$
17 $$1 + (-0.943 - 0.331i)T$$
19 $$1 + (-0.607 - 0.794i)T$$
23 $$1 + (0.748 - 0.663i)T$$
29 $$1 + (-0.527 - 0.849i)T$$
31 $$1 + (-0.215 - 0.976i)T$$
37 $$1 + (0.168 - 0.985i)T$$
41 $$1 + (0.607 + 0.794i)T$$
43 $$1 + (-0.120 + 0.992i)T$$
47 $$1 + (0.527 - 0.849i)T$$
53 $$1 + (-0.809 - 0.587i)T$$
59 $$1 + (-0.607 + 0.794i)T$$
61 $$1 + (0.809 - 0.587i)T$$
67 $$1 + (-0.120 - 0.992i)T$$
71 $$1 + (-0.995 - 0.0965i)T$$
73 $$1 + (0.309 + 0.951i)T$$
79 $$1 + (-0.681 + 0.732i)T$$
83 $$1 + (-0.443 + 0.896i)T$$
89 $$1 + T$$
97 $$1 + (0.262 - 0.964i)T$$
$$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$$