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Properties

Label 1-87-87.86-r1-0-0
Degree $1$
Conductor $87$
Sign $1$
Analytic cond. $9.34944$
Root an. cond. $9.34944$
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 + 4-s − 5-s + 7-s + 8-s − 10-s + 11-s + 13-s + 14-s + 16-s + 17-s − 19-s − 20-s + 22-s − 23-s + 25-s + 26-s + 28-s − 31-s + 32-s + 34-s − 35-s − 37-s − 38-s − 40-s + 41-s − 43-s + ⋯
L(s)  = 1  + 2-s + 4-s − 5-s + 7-s + 8-s − 10-s + 11-s + 13-s + 14-s + 16-s + 17-s − 19-s − 20-s + 22-s − 23-s + 25-s + 26-s + 28-s − 31-s + 32-s + 34-s − 35-s − 37-s − 38-s − 40-s + 41-s − 43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(87\)    =    \(3 \cdot 29\)
Sign: $1$
Analytic conductor: \(9.34944\)
Root analytic conductor: \(9.34944\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{87} (86, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 87,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.187069849\)
\(L(\frac12)\) \(\approx\) \(3.187069849\)
\(L(1)\) \(\approx\) \(2.020884517\)
\(L(1)\) \(\approx\) \(2.020884517\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
29 \( 1 \)
good2 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 + T \)
11 \( 1 + T \)
13 \( 1 + T \)
17 \( 1 + T \)
19 \( 1 - T \)
23 \( 1 - T \)
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

−30.41267943781182692509808742361, −29.901802554593012167546129129279, −28.08714605185508329151467838426, −27.5310399395176004346072989285, −25.9395405378404882275893640345, −24.78539794571253837300025026862, −23.761290743779364295623175032708, −23.13646303527543303570053239625, −21.88559974393812140543061382564, −20.80698782499690387080185267978, −19.906774299448537602928714359227, −18.71966272856570005576827149113, −17.0321041798943854803511841610, −15.926931445913140687475879233985, −14.821870219218430877480652409422, −14.06338154145621972708640835438, −12.47834150733352150379171651631, −11.60742102449555260890844925828, −10.71196680545714696593705399565, −8.50830069930215797153655224713, −7.399415894903088230289689512203, −5.974211343938127483361646786319, −4.45425895853171904044822515709, −3.56243286321819577554388853585, −1.54071818750898284645860140478, 1.54071818750898284645860140478, 3.56243286321819577554388853585, 4.45425895853171904044822515709, 5.974211343938127483361646786319, 7.399415894903088230289689512203, 8.50830069930215797153655224713, 10.71196680545714696593705399565, 11.60742102449555260890844925828, 12.47834150733352150379171651631, 14.06338154145621972708640835438, 14.821870219218430877480652409422, 15.926931445913140687475879233985, 17.0321041798943854803511841610, 18.71966272856570005576827149113, 19.906774299448537602928714359227, 20.80698782499690387080185267978, 21.88559974393812140543061382564, 23.13646303527543303570053239625, 23.761290743779364295623175032708, 24.78539794571253837300025026862, 25.9395405378404882275893640345, 27.5310399395176004346072989285, 28.08714605185508329151467838426, 29.901802554593012167546129129279, 30.41267943781182692509808742361

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