Properties

Degree 1
Conductor $ 2^{2} \cdot 3^{2} $
Sign $0.173 + 0.984i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)7-s + (0.5 + 0.866i)11-s + (−0.5 + 0.866i)13-s + 17-s − 19-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s + (−0.5 − 0.866i)29-s + (0.5 − 0.866i)31-s − 35-s + 37-s + (−0.5 + 0.866i)41-s + (0.5 + 0.866i)43-s + (0.5 + 0.866i)47-s + ⋯
L(s,χ)  = 1  + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)7-s + (0.5 + 0.866i)11-s + (−0.5 + 0.866i)13-s + 17-s − 19-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s + (−0.5 − 0.866i)29-s + (0.5 − 0.866i)31-s − 35-s + 37-s + (−0.5 + 0.866i)41-s + (0.5 + 0.866i)43-s + (0.5 + 0.866i)47-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.173 + 0.984i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 36 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.173 + 0.984i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(36\)    =    \(2^{2} \cdot 3^{2}\)
\( \varepsilon \)  =  $0.173 + 0.984i$
motivic weight  =  \(0\)
character  :  $\chi_{36} (7, \cdot )$
Sato-Tate  :  $\mu(6)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 36,\ (1:\ ),\ 0.173 + 0.984i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.9884724919 + 0.8294269034i$
$L(\frac12,\chi)$  $\approx$  $0.9884724919 + 0.8294269034i$
$L(\chi,1)$  $\approx$  0.9840438113 + 0.3581626565i
$L(1,\chi)$  $\approx$  0.9840438113 + 0.3581626565i

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

−35.29597037808016904534306015595, −34.106456299610876433114563561036, −32.603681757899809588003641453142, −31.87449655674035908979917366793, −30.31522251705824494748280115738, −29.33647098122860963978225052425, −27.637013665948661233708244495625, −27.12182721510882414006301090239, −25.31918922348646883417840897898, −24.09834201912689038847003266004, −23.19345118088844736555863741349, −21.459884940569361026427098634069, −20.26391341587413822319702352139, −19.23557391886182381803113946995, −17.34101325848358594101711625590, −16.46973826104502616682870882267, −14.85572121580761028053359824380, −13.38189167703740512806278827453, −11.991537439616035563313976849359, −10.54161693101067141682348472702, −8.72015991681402708810778978218, −7.46800994748448765693902597055, −5.32211014726077417771585399000, −3.74336585507794542185841000015, −0.93468572592624077763867749013, 2.36426765473592387475291733061, 4.395295916916735711983073054247, 6.4139388605395573100182027213, 7.90065445055288681844313381827, 9.61675299237772893847340912350, 11.29129554297552549381223229676, 12.35293697747913674603033103728, 14.48929246529189450685183375899, 15.14478864997353383792979024847, 16.93523973838671676594902327875, 18.43431621211412759036140745844, 19.32709737023475442080979923945, 21.03484422373811868900002898182, 22.24648206434101954039202024277, 23.38844143605730591265431483799, 24.827933827863368781211750821983, 26.04910307973596078760794988895, 27.34091390948471093109871095642, 28.33729745537479217726661183375, 29.9938104202461392898022778521, 30.90301731600950543144615679691, 31.974198935909409857618717886190, 33.72331345055919738890270307697, 34.38928170962187333286368319253, 35.65226917655706862275677463829

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