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Properties

Label 1-3e4-81.74-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: $\chi_{81} (74, \cdot )$
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

−30.98643488120245369056025227063, −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