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Properties

Label 1-3e4-81.2-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  + (0.993 + 0.116i)2-s + (0.973 + 0.230i)4-s + (−0.893 + 0.448i)5-s + (−0.286 + 0.957i)7-s + (0.939 + 0.342i)8-s + (−0.939 + 0.342i)10-s + (0.0581 + 0.998i)11-s + (0.597 + 0.802i)13-s + (−0.396 + 0.918i)14-s + (0.893 + 0.448i)16-s + (−0.766 − 0.642i)17-s + (0.766 − 0.642i)19-s + (−0.973 + 0.230i)20-s + (−0.0581 + 0.998i)22-s + (0.286 + 0.957i)23-s + ⋯
L(s)  = 1  + (0.993 + 0.116i)2-s + (0.973 + 0.230i)4-s + (−0.893 + 0.448i)5-s + (−0.286 + 0.957i)7-s + (0.939 + 0.342i)8-s + (−0.939 + 0.342i)10-s + (0.0581 + 0.998i)11-s + (0.597 + 0.802i)13-s + (−0.396 + 0.918i)14-s + (0.893 + 0.448i)16-s + (−0.766 − 0.642i)17-s + (0.766 − 0.642i)19-s + (−0.973 + 0.230i)20-s + (−0.0581 + 0.998i)22-s + (0.286 + 0.957i)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: $\chi_{81} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 81,\ (1:\ ),\ 0.0774 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.848843710 + 1.710716560i\)
\(L(\frac12)\) \(\approx\) \(1.848843710 + 1.710716560i\)
\(L(1)\) \(\approx\) \(1.583719281 + 0.6451750321i\)
\(L(1)\) \(\approx\) \(1.583719281 + 0.6451750321i\)

Euler product

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

−30.61732160071197634959115044702, −29.62827629096155646920515331992, −28.62640008449547450216894333140, −27.32395444338298043241523725259, −26.20134496227586264374269340131, −24.70808826309106549425186089432, −23.884689410550679585418797054524, −23.02148465984601937694208590542, −22.05455928864801591720946426943, −20.50513258777255337742176360230, −20.05984723004873252870623743130, −18.80265924307911258760513030274, −16.76858781866752450559198034765, −16.062965331509978304945252309575, −14.87783891408414156263988340088, −13.54200042207455506518458823916, −12.727439742853857203694108148242, −11.36070515787223163610291844654, −10.496372917462951089714435559516, −8.40126849345515229312689265224, −7.14432227340988677431811001221, −5.71757214299450011088698620105, −4.183469049991924482277258666254, −3.30285053937033776856161020056, −0.926066688338556489881723628874, 2.29940902779348274515744147746, 3.6848892326199020627740814617, 4.99592982556053768537266030424, 6.54477574649406860485817991392, 7.5360141378229731459374079026, 9.26364489073429849872837607201, 11.19072795335627318008666467043, 11.86969916014745243039454937512, 13.05998858621189422897052326201, 14.433955702377174041962718389957, 15.5082717229088148294804748126, 16.040646448909481252404601085571, 17.89701178951059577489916719702, 19.236984273318968566179848040873, 20.21974797166535372019214424060, 21.52371624961276425253480376478, 22.52120545584238444052120869884, 23.27375352019805296186151568228, 24.41238471994531314142451460630, 25.485968285938680184901628578635, 26.49239993547263245093972098447, 28.065345253125059032246062295877, 28.91058609975304961400573301946, 30.380902076955257586495721970224, 31.122191471746341447896499556287

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