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Properties

Label 1-87-87.38-r1-0-0
Degree $1$
Conductor $87$
Sign $0.00819 + 0.999i$
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  + (0.623 − 0.781i)2-s + (−0.222 − 0.974i)4-s + (−0.623 + 0.781i)5-s + (−0.222 + 0.974i)7-s + (−0.900 − 0.433i)8-s + (0.222 + 0.974i)10-s + (−0.900 + 0.433i)11-s + (−0.900 + 0.433i)13-s + (0.623 + 0.781i)14-s + (−0.900 + 0.433i)16-s + 17-s + (0.222 + 0.974i)19-s + (0.900 + 0.433i)20-s + (−0.222 + 0.974i)22-s + (−0.623 − 0.781i)23-s + ⋯
L(s)  = 1  + (0.623 − 0.781i)2-s + (−0.222 − 0.974i)4-s + (−0.623 + 0.781i)5-s + (−0.222 + 0.974i)7-s + (−0.900 − 0.433i)8-s + (0.222 + 0.974i)10-s + (−0.900 + 0.433i)11-s + (−0.900 + 0.433i)13-s + (0.623 + 0.781i)14-s + (−0.900 + 0.433i)16-s + 17-s + (0.222 + 0.974i)19-s + (0.900 + 0.433i)20-s + (−0.222 + 0.974i)22-s + (−0.623 − 0.781i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00819 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00819 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(87\)    =    \(3 \cdot 29\)
Sign: $0.00819 + 0.999i$
Analytic conductor: \(9.34944\)
Root analytic conductor: \(9.34944\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{87} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 87,\ (1:\ ),\ 0.00819 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5884051723 + 0.5932455688i\)
\(L(\frac12)\) \(\approx\) \(0.5884051723 + 0.5932455688i\)
\(L(1)\) \(\approx\) \(0.9496648704 - 0.07540864531i\)
\(L(1)\) \(\approx\) \(0.9496648704 - 0.07540864531i\)

Euler product

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

−30.24533117186404939578252653431, −29.31854060976622982790121368055, −27.76291023809591412926990613855, −26.7509584296775701993291824075, −25.86692813523053010300287028330, −24.50987538210153799952687264995, −23.74999872057572058864591657822, −23.02971699091162662529869972935, −21.6778777951714152909193937193, −20.56102038035737449212061427592, −19.542369684165137370614493148309, −17.81579712339528344374509198845, −16.68722219856535046248598035915, −16.00651337514499385449887351024, −14.804309897569043450698918526981, −13.45552312809108621277888573834, −12.70519613520848636041063077537, −11.385768809060424895830484652702, −9.62678955681992009080700193501, −7.98730900599210774708145448540, −7.392929168084611935796461789240, −5.5978979455021169735634604102, −4.50864109029888962495634049127, −3.20882554293817581576107533148, −0.302008017784760113629100837869, 2.228323118841654095255889641807, 3.317388569393063147831897015733, 4.88416489588722923179579792199, 6.21063546289647760161402657865, 7.82093823088393526100310949281, 9.60935925398030838155084738376, 10.597940149551576778749804011097, 11.97647533363562523202278858662, 12.525977756331641198519886816044, 14.26987855983460403606097046160, 14.98466386812304628360386675969, 16.1200497240408635823992223079, 18.27788201931554710740999034914, 18.79424724909622854060117983737, 19.89701950490707829627785578532, 21.165881381354207934509804803161, 22.114576214326998692354314387646, 22.98272425293331146362696627486, 23.93850607794220373334217165696, 25.26613626539109291649083410003, 26.63767526362683146136114643184, 27.698202626229247234692285546719, 28.7060780911719942739223504668, 29.63560627863646152078599759978, 30.85024098517124743316384607318

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