Properties

Degree 1
Conductor $ 3^{2} \cdot 13 $
Sign $-0.568 - 0.822i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)4-s + (−0.5 − 0.866i)5-s + 7-s + 8-s + (−0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s + (−0.5 − 0.866i)14-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s + 20-s + (−0.5 + 0.866i)22-s + 23-s + (−0.5 + 0.866i)25-s + ⋯
L(s,χ)  = 1  + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)4-s + (−0.5 − 0.866i)5-s + 7-s + 8-s + (−0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s + (−0.5 − 0.866i)14-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s + 20-s + (−0.5 + 0.866i)22-s + 23-s + (−0.5 + 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.568 - 0.822i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 117 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.568 - 0.822i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(117\)    =    \(3^{2} \cdot 13\)
\( \varepsilon \)  =  $-0.568 - 0.822i$
motivic weight  =  \(0\)
character  :  $\chi_{117} (94, \cdot )$
Sato-Tate  :  $\mu(3)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 117,\ (0:\ ),\ -0.568 - 0.822i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.3272020685 - 0.6236595103i$
$L(\frac12,\chi)$  $\approx$  $0.3272020685 - 0.6236595103i$
$L(\chi,1)$  $\approx$  0.6092307767 - 0.4523545595i
$L(1,\chi)$  $\approx$  0.6092307767 - 0.4523545595i

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

−29.476128798765513316300051459494, −28.1222743243577828787730216397, −27.39331885782154464021542621893, −26.49147462859656884573817179771, −25.65466844907062860913250140975, −24.532850514261164074376792056955, −23.48142470096563050658433857212, −22.868027896841967302355729746197, −21.47146434636615195533983101330, −20.06613749201794607388649960323, −18.94817497813068937807513392252, −18.05399330274133466195607137024, −17.28090394576170395079731221581, −15.87248450927147953618987043722, −14.85942618271083343781708662694, −14.405076242345922528066136363311, −12.73418998146692026794891117218, −11.04309202868233221920874749066, −10.35931792492447020530943783194, −8.760443708961488373127537721873, −7.70525068946875589390460985472, −6.87071377532156804539725721223, −5.39227524146992959741420388045, −4.09473606683933394613154061229, −1.9217459542032920769503885692, 0.84271746533863841867246822597, 2.51158245168858100169272620874, 4.166435200581174912962655160254, 5.17394777025243957520750536029, 7.491820926453985660084735130061, 8.453902926561549144785591532344, 9.29429105054539043650417439331, 11.01722856288380749758648775171, 11.49274573224915293153329731611, 12.8378939261710372050163417897, 13.7233082045561054701219963549, 15.4127051888604724876241617631, 16.63393726936574369524251789728, 17.52771918362073585715779988323, 18.66326969749435446255521106329, 19.63128596038387446901213162145, 20.75574199982465845798076006190, 21.19839870716268253861305875512, 22.569315367038952528550131905979, 23.87803974662025539591103006692, 24.68599351890544203885352878326, 26.17378464351798208462444718835, 27.1966623992246691089933557675, 27.76912914263876936379408119667, 28.80608126597129297283194772803

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