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 $0$

Origins

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Normalization:  

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

Graph of the $Z$-function along the critical line