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Properties

Label 1-87-87.44-r0-0-0
Degree $1$
Conductor $87$
Sign $-0.347 - 0.937i$
Analytic cond. $0.404026$
Root an. cond. $0.404026$
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  + (−0.974 + 0.222i)2-s + (0.900 − 0.433i)4-s + (−0.222 − 0.974i)5-s + (−0.900 − 0.433i)7-s + (−0.781 + 0.623i)8-s + (0.433 + 0.900i)10-s + (−0.781 − 0.623i)11-s + (−0.623 + 0.781i)13-s + (0.974 + 0.222i)14-s + (0.623 − 0.781i)16-s − i·17-s + (−0.433 − 0.900i)19-s + (−0.623 − 0.781i)20-s + (0.900 + 0.433i)22-s + (0.222 − 0.974i)23-s + ⋯
L(s)  = 1  + (−0.974 + 0.222i)2-s + (0.900 − 0.433i)4-s + (−0.222 − 0.974i)5-s + (−0.900 − 0.433i)7-s + (−0.781 + 0.623i)8-s + (0.433 + 0.900i)10-s + (−0.781 − 0.623i)11-s + (−0.623 + 0.781i)13-s + (0.974 + 0.222i)14-s + (0.623 − 0.781i)16-s − i·17-s + (−0.433 − 0.900i)19-s + (−0.623 − 0.781i)20-s + (0.900 + 0.433i)22-s + (0.222 − 0.974i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.347 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.347 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(87\)    =    \(3 \cdot 29\)
Sign: $-0.347 - 0.937i$
Analytic conductor: \(0.404026\)
Root analytic conductor: \(0.404026\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{87} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 87,\ (0:\ ),\ -0.347 - 0.937i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2256576882 - 0.3241404843i\)
\(L(\frac12)\) \(\approx\) \(0.2256576882 - 0.3241404843i\)
\(L(1)\) \(\approx\) \(0.4983785848 - 0.1668469250i\)
\(L(1)\) \(\approx\) \(0.4983785848 - 0.1668469250i\)

Euler product

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

−30.698037830858702148361992759465, −29.70056809401672360221927617141, −28.848740918269470942992929415686, −27.75755787930730019019203827991, −26.72641394290546500275123599024, −25.796992555553328230123590373282, −25.15629255921281608152665203005, −23.46912822349494809232298647853, −22.325273446826651860058968928301, −21.26180560420405755727427210140, −19.84080332389092549699537442166, −19.06321597025369347829347093321, −18.14905668364804599604907251480, −17.06973613252476247206952671754, −15.60816391846011196135664488171, −15.01792721272147053140431260601, −12.95931176154838629305780738570, −11.92509955063689648790635170300, −10.42541620782639787122352893585, −9.91030253394485258906195484161, −8.233712988810272549001364539954, −7.16603367802219739860781976106, −5.951123668852292554795815955705, −3.43336222746076030937601919125, −2.31541580346449635764004829475, 0.5460702679273827689360969974, 2.67569610231573472112056643611, 4.771555465662258450734495449518, 6.37698827508419098696622601900, 7.61709395956936408435774582461, 8.880556774481200832737203562522, 9.76914340293982539970581223706, 11.13125802403490834046341884081, 12.427850514509524640249895786097, 13.72260344062817033469926269535, 15.472991180713704129830744157135, 16.38429156064819883743410498019, 17.00103123152823605659885265998, 18.50942746966400294254481407656, 19.49266016495678401205602756524, 20.333464912561036855024808711111, 21.46347612207348430665650149101, 23.23914222878089916059319820390, 24.16240399781157405785962432373, 25.07609356065097860166020703035, 26.348345501876655879661267542954, 26.95462010578207472177478485767, 28.402766543534513099440064168405, 28.883093710304518904211975088213, 29.90657896017135812206658938365

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