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Properties

Label 1-3e4-81.56-r1-0-0
Degree $1$
Conductor $81$
Sign $0.824 - 0.565i$
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  + (−0.597 + 0.802i)2-s + (−0.286 − 0.957i)4-s + (0.835 + 0.549i)5-s + (−0.686 − 0.727i)7-s + (0.939 + 0.342i)8-s + (−0.939 + 0.342i)10-s + (−0.893 − 0.448i)11-s + (0.396 − 0.918i)13-s + (0.993 − 0.116i)14-s + (−0.835 + 0.549i)16-s + (−0.766 − 0.642i)17-s + (0.766 − 0.642i)19-s + (0.286 − 0.957i)20-s + (0.893 − 0.448i)22-s + (0.686 − 0.727i)23-s + ⋯
L(s)  = 1  + (−0.597 + 0.802i)2-s + (−0.286 − 0.957i)4-s + (0.835 + 0.549i)5-s + (−0.686 − 0.727i)7-s + (0.939 + 0.342i)8-s + (−0.939 + 0.342i)10-s + (−0.893 − 0.448i)11-s + (0.396 − 0.918i)13-s + (0.993 − 0.116i)14-s + (−0.835 + 0.549i)16-s + (−0.766 − 0.642i)17-s + (0.766 − 0.642i)19-s + (0.286 − 0.957i)20-s + (0.893 − 0.448i)22-s + (0.686 − 0.727i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.824 - 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.824 - 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(81\)    =    \(3^{4}\)
Sign: $0.824 - 0.565i$
Analytic conductor: \(8.70465\)
Root analytic conductor: \(8.70465\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{81} (56, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 81,\ (1:\ ),\ 0.824 - 0.565i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9760299288 - 0.3025466108i\)
\(L(\frac12)\) \(\approx\) \(0.9760299288 - 0.3025466108i\)
\(L(1)\) \(\approx\) \(0.8079441753 + 0.07197521243i\)
\(L(1)\) \(\approx\) \(0.8079441753 + 0.07197521243i\)

Euler product

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

−30.89310852338307113948718023381, −29.283562240495709088682048521451, −28.77688474477087338887608115724, −28.08692634695840246613963768419, −26.51199410488068123239624102965, −25.7226496993346751193287965480, −24.75279172770236171880721113663, −23.08708291159586995461294210357, −21.741903929909234895339865303299, −21.14821910864311992802782688073, −19.98237682009209999941887068652, −18.78002238721492769097944655769, −17.90306449617602486728740697731, −16.75132118564639655137684867754, −15.65641513437985504915760064119, −13.620657539978405722283504461513, −12.82857755815031678507427159084, −11.69810466184592313815299820453, −10.14987481997479060559164892399, −9.34232808102542672851006307127, −8.24718156052587009206979184742, −6.408036659739737929809163459415, −4.75632755644399388928709779207, −2.89286037731367279932198220096, −1.57357992113764614169910904856, 0.61684017480868729454737131053, 2.84394147514700377485976065657, 5.10118949560273703843638177535, 6.36143218570510640053509928465, 7.34002926327895227342633471542, 8.84865621804764099853288079429, 10.14416526923966496518898754766, 10.801563373123930192418421353665, 13.27021246328645089972948921031, 13.862211143333380928365328818428, 15.37388123011043664227372819935, 16.28922451668678074340387688471, 17.58405527164533911352253852073, 18.27108135323765144949354798261, 19.506557378882574131707678850, 20.73279498072467112898452206828, 22.37906745305228947569484813614, 23.12069585337849489293225532565, 24.460467170939492778447944503365, 25.4554003166452978293543696038, 26.36646503719704713137657689030, 26.99384793893313949832167616244, 28.6959252748132902641341021272, 29.19361747689156514954786803495, 30.55463942478311568961383795761

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