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Properties

Label 1-3e4-81.23-r1-0-0
Degree $1$
Conductor $81$
Sign $0.0774 - 0.996i$
Analytic cond. $8.70465$
Root an. cond. $8.70465$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(81\)    =    \(3^{4}\)
Sign: $0.0774 - 0.996i$
Analytic conductor: \(8.70465\)
Root analytic conductor: \(8.70465\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{81} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 81,\ (1:\ ),\ 0.0774 - 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3633567290 - 0.3362103408i\)
\(L(\frac12)\) \(\approx\) \(0.3633567290 - 0.3362103408i\)
\(L(1)\) \(\approx\) \(0.7503756943 + 0.1887371641i\)
\(L(1)\) \(\approx\) \(0.7503756943 + 0.1887371641i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 + (0.286 + 0.957i)T \)
5 \( 1 + (-0.396 - 0.918i)T \)
7 \( 1 + (-0.0581 + 0.998i)T \)
11 \( 1 + (-0.597 - 0.802i)T \)
13 \( 1 + (-0.686 - 0.727i)T \)
17 \( 1 + (-0.173 - 0.984i)T \)
19 \( 1 + (0.173 - 0.984i)T \)
23 \( 1 + (0.0581 + 0.998i)T \)
29 \( 1 + (-0.973 - 0.230i)T \)
31 \( 1 + (0.893 + 0.448i)T \)
37 \( 1 + (-0.939 - 0.342i)T \)
41 \( 1 + (0.286 - 0.957i)T \)
43 \( 1 + (-0.993 + 0.116i)T \)
47 \( 1 + (-0.893 + 0.448i)T \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 + (-0.597 + 0.802i)T \)
61 \( 1 + (-0.835 - 0.549i)T \)
67 \( 1 + (0.973 - 0.230i)T \)
71 \( 1 + (-0.766 + 0.642i)T \)
73 \( 1 + (0.766 + 0.642i)T \)
79 \( 1 + (-0.286 - 0.957i)T \)
83 \( 1 + (0.286 + 0.957i)T \)
89 \( 1 + (-0.766 - 0.642i)T \)
97 \( 1 + (0.396 - 0.918i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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