Properties

Degree 1
Conductor $ 3 \cdot 29 $
Sign $-0.347 + 0.937i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(87\)    =    \(3 \cdot 29\)
\( \varepsilon \)  =  $-0.347 + 0.937i$
motivic weight  =  \(0\)
character  :  $\chi_{87} (2, \cdot )$
Sato-Tate  :  $\mu(28)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 87,\ (0:\ ),\ -0.347 + 0.937i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.2256576882 + 0.3241404843i$
$L(\frac12,\chi)$  $\approx$  $0.2256576882 + 0.3241404843i$
$L(\chi,1)$  $\approx$  0.4983785848 + 0.1668469250i
$L(1,\chi)$  $\approx$  0.4983785848 + 0.1668469250i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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