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Properties

Label 1-87-87.53-r1-0-0
Degree $1$
Conductor $87$
Sign $-0.510 + 0.859i$
Analytic cond. $9.34944$
Root an. cond. $9.34944$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.222 − 0.974i)2-s + (−0.900 − 0.433i)4-s + (0.222 − 0.974i)5-s + (−0.900 + 0.433i)7-s + (−0.623 + 0.781i)8-s + (−0.900 − 0.433i)10-s + (−0.623 − 0.781i)11-s + (0.623 + 0.781i)13-s + (0.222 + 0.974i)14-s + (0.623 + 0.781i)16-s − 17-s + (−0.900 − 0.433i)19-s + (−0.623 + 0.781i)20-s + (−0.900 + 0.433i)22-s + (0.222 + 0.974i)23-s + ⋯
L(s)  = 1  + (0.222 − 0.974i)2-s + (−0.900 − 0.433i)4-s + (0.222 − 0.974i)5-s + (−0.900 + 0.433i)7-s + (−0.623 + 0.781i)8-s + (−0.900 − 0.433i)10-s + (−0.623 − 0.781i)11-s + (0.623 + 0.781i)13-s + (0.222 + 0.974i)14-s + (0.623 + 0.781i)16-s − 17-s + (−0.900 − 0.433i)19-s + (−0.623 + 0.781i)20-s + (−0.900 + 0.433i)22-s + (0.222 + 0.974i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.510 + 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.510 + 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(87\)    =    \(3 \cdot 29\)
Sign: $-0.510 + 0.859i$
Analytic conductor: \(9.34944\)
Root analytic conductor: \(9.34944\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{87} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 87,\ (1:\ ),\ -0.510 + 0.859i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2160583614 - 0.3797082842i\)
\(L(\frac12)\) \(\approx\) \(-0.2160583614 - 0.3797082842i\)
\(L(1)\) \(\approx\) \(0.5729372282 - 0.5053632141i\)
\(L(1)\) \(\approx\) \(0.5729372282 - 0.5053632141i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.222 - 0.974i)T \)
5 \( 1 + (0.222 - 0.974i)T \)
7 \( 1 + (-0.900 + 0.433i)T \)
11 \( 1 + (-0.623 - 0.781i)T \)
13 \( 1 + (0.623 + 0.781i)T \)
17 \( 1 - T \)
19 \( 1 + (-0.900 - 0.433i)T \)
23 \( 1 + (0.222 + 0.974i)T \)
31 \( 1 + (-0.222 + 0.974i)T \)
37 \( 1 + (0.623 - 0.781i)T \)
41 \( 1 - T \)
43 \( 1 + (-0.222 - 0.974i)T \)
47 \( 1 + (-0.623 - 0.781i)T \)
53 \( 1 + (0.222 - 0.974i)T \)
59 \( 1 - T \)
61 \( 1 + (-0.900 + 0.433i)T \)
67 \( 1 + (0.623 - 0.781i)T \)
71 \( 1 + (-0.623 - 0.781i)T \)
73 \( 1 + (-0.222 - 0.974i)T \)
79 \( 1 + (0.623 - 0.781i)T \)
83 \( 1 + (0.900 + 0.433i)T \)
89 \( 1 + (0.222 - 0.974i)T \)
97 \( 1 + (-0.900 - 0.433i)T \)
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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

−31.11175260202981977219372692871, −30.282040821692499342158369878034, −29.05771135935434713241573472815, −27.647398599461680446922001472, −26.3908356035690106153040601309, −25.890149251925311851209432010251, −24.94949179744574170358774986903, −23.37986438488238240543348225659, −22.8161621735661490920906140600, −21.92563614438512350676398445535, −20.41790936423356876128597999735, −18.843415638047952396035209738867, −18.00091992539438771027040843460, −16.88622101448467361903776833394, −15.593180572387328698609214692564, −14.86813428322236336222410013967, −13.486778181799888868823250465227, −12.760085875264573407322190044255, −10.70812455518090735850057547915, −9.68497107129613431545879871950, −8.08430770416771802990306505691, −6.84736736531837968725715422341, −6.05838444924759586600520392278, −4.30566745982696447720399072742, −2.88121892079078274310620245184, 0.18063263501717368619651199965, 1.973522235697160544174745693620, 3.54729376055463379390397916885, 4.97831101171918519933446383325, 6.21691304335012414158352841875, 8.63144081877288052631931824290, 9.242682643766734600969232837659, 10.6834811795253417883990433649, 11.902078328319829117615574777303, 13.1027969802638298721381989035, 13.58749218066166171958421445966, 15.451868823934733281383511883615, 16.59129445388832498557193539650, 17.99039529506027631006069597288, 19.15097761562047262851957682913, 19.958458160078495555671823709939, 21.28361724632152800650698861100, 21.76099813028790679680593395824, 23.27336807538897929376802865121, 24.070471673391629408173256863772, 25.50635654471801026202863895179, 26.69800812648269484808578754522, 27.99343680320556129593449376935, 28.81923347707959056877739384268, 29.31805772955709965394669169345

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