Properties

Degree 1
Conductor $ 2^{2} \cdot 23 $
Sign $-0.392 - 0.919i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.841 − 0.540i)3-s + (0.415 − 0.909i)5-s + (0.959 + 0.281i)7-s + (0.415 + 0.909i)9-s + (0.654 − 0.755i)11-s + (−0.959 + 0.281i)13-s + (−0.841 + 0.540i)15-s + (−0.142 − 0.989i)17-s + (0.142 − 0.989i)19-s + (−0.654 − 0.755i)21-s + (−0.654 − 0.755i)25-s + (0.142 − 0.989i)27-s + (−0.142 − 0.989i)29-s + (−0.841 + 0.540i)31-s + (−0.959 + 0.281i)33-s + ⋯
L(s,χ)  = 1  + (−0.841 − 0.540i)3-s + (0.415 − 0.909i)5-s + (0.959 + 0.281i)7-s + (0.415 + 0.909i)9-s + (0.654 − 0.755i)11-s + (−0.959 + 0.281i)13-s + (−0.841 + 0.540i)15-s + (−0.142 − 0.989i)17-s + (0.142 − 0.989i)19-s + (−0.654 − 0.755i)21-s + (−0.654 − 0.755i)25-s + (0.142 − 0.989i)27-s + (−0.142 − 0.989i)29-s + (−0.841 + 0.540i)31-s + (−0.959 + 0.281i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.392 - 0.919i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 92 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.392 - 0.919i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(92\)    =    \(2^{2} \cdot 23\)
\( \varepsilon \)  =  $-0.392 - 0.919i$
motivic weight  =  \(0\)
character  :  $\chi_{92} (75, \cdot )$
Sato-Tate  :  $\mu(22)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 92,\ (1:\ ),\ -0.392 - 0.919i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.7111909106 - 1.076729550i$
$L(\frac12,\chi)$  $\approx$  $0.7111909106 - 1.076729550i$
$L(\chi,1)$  $\approx$  0.8628975937 - 0.4210481654i
$L(1,\chi)$  $\approx$  0.8628975937 - 0.4210481654i

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

−30.1525683146824678227463085157, −29.5256242921364533651667027065, −28.21938888434475341821211739431, −27.27248269180289795591194532442, −26.54819565859175912219021329954, −25.208727876736604418699565770561, −23.94842494662172116922687374776, −22.85219433629511835448746630118, −22.02622087888949005983536434520, −21.161186460963174116872997545360, −19.86703926308540094621746252671, −18.23978053377389612595622251616, −17.54301399030236973431003812620, −16.649986022342650632012977709538, −14.84037501363039116968190860418, −14.6214660379216703616916134108, −12.64306980516852390151225968573, −11.43622514152918811879244828132, −10.49730682232510423790731945202, −9.569721712563948454136586678601, −7.61646625995718237255854537494, −6.40186722886582272152100549114, −5.11601592361277779822024887575, −3.80991530204000556504147132120, −1.74326981650460929127055213301, 0.675585747098607964481219342732, 2.07388394478194674423780893993, 4.68120245092933230636497717886, 5.46248851137550244832340106086, 6.91484084776923333973023220680, 8.31103128943902432729595089774, 9.569958817076489571476168376237, 11.29912879065779390543300740327, 11.94523330617818750317027361140, 13.21848910966914142698458696989, 14.26585914033969274248888660519, 15.97344549447729242779332008316, 17.05902208392097799704809704469, 17.682005704791671388645428867022, 18.92245498673701864873288691373, 20.17991571993627594194857762817, 21.501599054232700947390957466169, 22.21280000119547995127120368485, 23.79339508778249633081551960055, 24.42363049952314867664858846788, 25.07662948554512841904379044696, 26.994458306924141334277006690530, 27.7908425781104068294759989373, 28.8162605501141968180309984924, 29.58060805391190405893378701819

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