# Properties

 Label 1-281-281.280-r0-0-0 Degree $1$ Conductor $281$ Sign $1$ Analytic cond. $1.30495$ Root an. cond. $1.30495$ Motivic weight $0$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + 2-s − 3-s + 4-s + 5-s − 6-s + 7-s + 8-s + 9-s + 10-s − 11-s − 12-s − 13-s + 14-s − 15-s + 16-s + 17-s + 18-s − 19-s + 20-s − 21-s − 22-s − 23-s − 24-s + 25-s − 26-s − 27-s + 28-s + ⋯
 L(s)  = 1 + 2-s − 3-s + 4-s + 5-s − 6-s + 7-s + 8-s + 9-s + 10-s − 11-s − 12-s − 13-s + 14-s − 15-s + 16-s + 17-s + 18-s − 19-s + 20-s − 21-s − 22-s − 23-s − 24-s + 25-s − 26-s − 27-s + 28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 281 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 281 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$281$$ Sign: $1$ Analytic conductor: $$1.30495$$ Root analytic conductor: $$1.30495$$ Motivic weight: $$0$$ Rational: yes Arithmetic: yes Character: $\chi_{281} (280, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(1,\ 281,\ (0:\ ),\ 1)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$2.131139298$$ $$L(\frac12)$$ $$\approx$$ $$2.131139298$$ $$L(1)$$ $$\approx$$ $$1.738376867$$ $$L(1)$$ $$\approx$$ $$1.738376867$$

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad281 $$1$$
good2 $$1 + T$$
3 $$1 - T$$
5 $$1 + T$$
7 $$1 + T$$
11 $$1 - T$$
13 $$1 - T$$
17 $$1 + T$$
19 $$1 - T$$
23 $$1 - T$$
29 $$1 + T$$
31 $$1 + T$$
37 $$1 - T$$
41 $$1 - T$$
43 $$1 + T$$
47 $$1 - T$$
53 $$1 + T$$
59 $$1 + T$$
61 $$1 - T$$
67 $$1 - T$$
71 $$1 - T$$
73 $$1 - T$$
79 $$1 + T$$
83 $$1 - T$$
89 $$1 - T$$
97 $$1 - T$$
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$$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−25.29354639279870071744652278608, −24.36419550991184813998164672727, −23.80221534264487579729065373573, −22.89962854872351366554077634406, −21.91295786256722900220782941699, −21.23868678608654432894892755950, −20.80717877064518640692772736075, −19.21956559306339212756214051370, −18.00058118851863777767600799839, −17.28245669114536621956282860103, −16.45473732154936107936477228719, −15.32174637059639457944894828384, −14.37974884890450449782829788729, −13.48333398421327876962202960186, −12.441339902189964110056658285162, −11.800663499549504357679756724147, −10.44791931157495323571293912530, −10.19329290466318467332665283777, −8.09396816489860938252231503331, −6.995757926726105086787928902398, −5.88664777401239263400143987243, −5.17976060191625675917745865828, −4.449085683198577513178564600393, −2.58683816442763476633633732272, −1.55149650239701073582208321398, 1.55149650239701073582208321398, 2.58683816442763476633633732272, 4.449085683198577513178564600393, 5.17976060191625675917745865828, 5.88664777401239263400143987243, 6.995757926726105086787928902398, 8.09396816489860938252231503331, 10.19329290466318467332665283777, 10.44791931157495323571293912530, 11.800663499549504357679756724147, 12.441339902189964110056658285162, 13.48333398421327876962202960186, 14.37974884890450449782829788729, 15.32174637059639457944894828384, 16.45473732154936107936477228719, 17.28245669114536621956282860103, 18.00058118851863777767600799839, 19.21956559306339212756214051370, 20.80717877064518640692772736075, 21.23868678608654432894892755950, 21.91295786256722900220782941699, 22.89962854872351366554077634406, 23.80221534264487579729065373573, 24.36419550991184813998164672727, 25.29354639279870071744652278608