Properties

Degree 1
Conductor $ 3^{4} $
Sign $0.0774 + 0.996i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.286 − 0.957i)2-s + (−0.835 − 0.549i)4-s + (−0.396 + 0.918i)5-s + (−0.0581 − 0.998i)7-s + (−0.766 + 0.642i)8-s + (0.766 + 0.642i)10-s + (−0.597 + 0.802i)11-s + (−0.686 + 0.727i)13-s + (−0.973 − 0.230i)14-s + (0.396 + 0.918i)16-s + (−0.173 + 0.984i)17-s + (0.173 + 0.984i)19-s + (0.835 − 0.549i)20-s + (0.597 + 0.802i)22-s + (0.0581 − 0.998i)23-s + ⋯
L(s,χ)  = 1  + (0.286 − 0.957i)2-s + (−0.835 − 0.549i)4-s + (−0.396 + 0.918i)5-s + (−0.0581 − 0.998i)7-s + (−0.766 + 0.642i)8-s + (0.766 + 0.642i)10-s + (−0.597 + 0.802i)11-s + (−0.686 + 0.727i)13-s + (−0.973 − 0.230i)14-s + (0.396 + 0.918i)16-s + (−0.173 + 0.984i)17-s + (0.173 + 0.984i)19-s + (0.835 − 0.549i)20-s + (0.597 + 0.802i)22-s + (0.0581 − 0.998i)23-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.0774 + 0.996i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 81 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.0774 + 0.996i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(81\)    =    \(3^{4}\)
\( \varepsilon \)  =  $0.0774 + 0.996i$
motivic weight  =  \(0\)
character  :  $\chi_{81} (74, \cdot )$
Sato-Tate  :  $\mu(54)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 81,\ (1:\ ),\ 0.0774 + 0.996i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.3633567290 + 0.3362103408i$
$L(\frac12,\chi)$  $\approx$  $0.3633567290 + 0.3362103408i$
$L(\chi,1)$  $\approx$  0.7503756943 - 0.1887371641i
$L(1,\chi)$  $\approx$  0.7503756943 - 0.1887371641i

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

−30.98643488120245369056025227063, −29.40119005333053009929463990005, −28.075956980612592163331993380, −27.267575932418893064730758575676, −26.07276812625422679067416840945, −24.72288504158318736821455545447, −24.41553581495347872145209722164, −23.12871578365785566535057565866, −21.99085356037206344837254490961, −21.00213821187028607593779073953, −19.46785013976502761027485779497, −18.20699500632264013620279377098, −17.097486041989516456339537228311, −15.81458540547049414045884208205, −15.39229807930233114643092781769, −13.72020782899272261220099578299, −12.73601158967182409515282805973, −11.649834833079240368025539153970, −9.44138874578974627967751968747, −8.51591018667244545920768409184, −7.40408799169888742101201837272, −5.62550559540942635239908718588, −4.91813922531657441151025791821, −3.07473948499261211020509268739, −0.20894345207890676410712156456, 1.98700699059944945287562399711, 3.53392071510175324621183838193, 4.63393008637536657576843040671, 6.57728561083320115398703670743, 7.970114596213606310471246670196, 9.91414622773974632368774094115, 10.5558880427903700859274415864, 11.78197205279547136791329201342, 12.97964190716918769042229298794, 14.25337508018663507085796480838, 15.04797968744190793917806253733, 16.89616732619248716850099817540, 18.19032958418672018386178912474, 19.16547148652299215522513793255, 20.1200526164494567599861360473, 21.18379277870326013234972520157, 22.469084751259076640868618336305, 23.13745133431564020362358495888, 24.13848557809094004075305098529, 26.21756836357427736211656809849, 26.674466016983443544668596711685, 27.94436592169430128127121460671, 29.09122554602857686880188750750, 29.997129907465081206032553615068, 30.83694305485555984690907010781

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