Properties

Degree 1
Conductor $ 3^{4} $
Sign $-0.431 - 0.902i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.396 − 0.918i)2-s + (−0.686 − 0.727i)4-s + (−0.0581 − 0.998i)5-s + (0.973 + 0.230i)7-s + (−0.939 + 0.342i)8-s + (−0.939 − 0.342i)10-s + (−0.835 − 0.549i)11-s + (−0.993 − 0.116i)13-s + (0.597 − 0.802i)14-s + (−0.0581 + 0.998i)16-s + (0.766 − 0.642i)17-s + (0.766 + 0.642i)19-s + (−0.686 + 0.727i)20-s + (−0.835 + 0.549i)22-s + (0.973 − 0.230i)23-s + ⋯
L(s,χ)  = 1  + (0.396 − 0.918i)2-s + (−0.686 − 0.727i)4-s + (−0.0581 − 0.998i)5-s + (0.973 + 0.230i)7-s + (−0.939 + 0.342i)8-s + (−0.939 − 0.342i)10-s + (−0.835 − 0.549i)11-s + (−0.993 − 0.116i)13-s + (0.597 − 0.802i)14-s + (−0.0581 + 0.998i)16-s + (0.766 − 0.642i)17-s + (0.766 + 0.642i)19-s + (−0.686 + 0.727i)20-s + (−0.835 + 0.549i)22-s + (0.973 − 0.230i)23-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.431 - 0.902i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 81 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.431 - 0.902i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(81\)    =    \(3^{4}\)
\( \varepsilon \)  =  $-0.431 - 0.902i$
motivic weight  =  \(0\)
character  :  $\chi_{81} (67, \cdot )$
Sato-Tate  :  $\mu(27)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 81,\ (0:\ ),\ -0.431 - 0.902i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.5805677255 - 0.9211334058i$
$L(\frac12,\chi)$  $\approx$  $0.5805677255 - 0.9211334058i$
$L(\chi,1)$  $\approx$  0.8926723115 - 0.7319929517i
$L(1,\chi)$  $\approx$  0.8926723115 - 0.7319929517i

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

−31.16440001504374770668781967300, −30.69211531041409843287706384201, −29.53098147443102754339180508817, −27.81135626460366291094329222837, −26.73147400794318134797634915840, −26.10626023665248757243986982186, −24.85477770218605740755474225720, −23.79061620509221527870549591945, −22.94890467788409386017591278041, −21.832733022559817100432300079102, −20.8760522349453240314670263372, −19.084270110311622593594586770833, −17.893299812440918145257983975769, −17.19882557004835888362721675830, −15.57138249958401774187701377616, −14.75923869356930386827057302162, −13.915336310002339488803776517024, −12.476838208515292456804713355420, −11.07908240157990253125123494155, −9.63605251108097479916600461823, −7.80136252249749110460619682494, −7.26875446176399156475584938155, −5.63248459346330518422203139038, −4.40426135896923152165104539105, −2.71521232458012014816122388264, 1.270726801549941400468941468914, 2.95926889257868604527613272598, 4.826828945625309867916295877378, 5.37187700050843853542629625944, 7.87902985260329981942691539816, 9.060642767295216046412595967785, 10.36353543065501368868515054165, 11.70708652020958162340456988341, 12.50980441473130305683360228944, 13.753104740832762801251271961775, 14.831772070574937833709673213917, 16.33510027859512951277543783493, 17.73989043141732280797792133871, 18.79224700437183898873861161239, 20.091025804704880097734999405928, 20.92982157560949932083919966266, 21.66125269361211144000942772639, 23.14681543960622161365755425551, 24.10200418144850302103961218607, 24.925658299454634550166162502302, 27.00931690687988539602988849981, 27.50470127721742863225583440292, 28.82029616803557596077859696109, 29.37636447890178029198875258853, 30.860557697265767055381023345811

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