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Properties

Label 1-3e4-81.4-r0-0-0
Degree $1$
Conductor $81$
Sign $0.996 + 0.0774i$
Analytic cond. $0.376162$
Root an. cond. $0.376162$
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  + (0.973 + 0.230i)2-s + (0.893 + 0.448i)4-s + (0.597 − 0.802i)5-s + (−0.835 − 0.549i)7-s + (0.766 + 0.642i)8-s + (0.766 − 0.642i)10-s + (−0.993 + 0.116i)11-s + (−0.286 + 0.957i)13-s + (−0.686 − 0.727i)14-s + (0.597 + 0.802i)16-s + (0.173 + 0.984i)17-s + (0.173 − 0.984i)19-s + (0.893 − 0.448i)20-s + (−0.993 − 0.116i)22-s + (−0.835 + 0.549i)23-s + ⋯
L(s)  = 1  + (0.973 + 0.230i)2-s + (0.893 + 0.448i)4-s + (0.597 − 0.802i)5-s + (−0.835 − 0.549i)7-s + (0.766 + 0.642i)8-s + (0.766 − 0.642i)10-s + (−0.993 + 0.116i)11-s + (−0.286 + 0.957i)13-s + (−0.686 − 0.727i)14-s + (0.597 + 0.802i)16-s + (0.173 + 0.984i)17-s + (0.173 − 0.984i)19-s + (0.893 − 0.448i)20-s + (−0.993 − 0.116i)22-s + (−0.835 + 0.549i)23-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(81\)    =    \(3^{4}\)
Sign: $0.996 + 0.0774i$
Analytic conductor: \(0.376162\)
Root analytic conductor: \(0.376162\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{81} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 81,\ (0:\ ),\ 0.996 + 0.0774i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.623567761 + 0.06300182309i\)
\(L(\frac12)\) \(\approx\) \(1.623567761 + 0.06300182309i\)
\(L(1)\) \(\approx\) \(1.650399504 + 0.07156843997i\)
\(L(1)\) \(\approx\) \(1.650399504 + 0.07156843997i\)

Euler product

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

−31.13215627610484408298584692250, −29.77157615615157276523676460043, −29.30337781833074885468723731493, −28.181415651877560799685829836343, −26.487783442083226522231887751625, −25.40669191423393596509449772985, −24.64743582634621786772682906209, −22.991359635490563353887072326455, −22.5202905178582941041306129857, −21.438792009116496059367427289909, −20.41601228867488936067057927224, −19.03448940590677206871190020674, −18.11488496943427504281887082373, −16.271207350810522720602246147895, −15.33428513180148347774956452463, −14.18682295174626141009446036693, −13.139073466883395744639931467813, −12.11581489157847531618208051144, −10.60030673078252877937333771063, −9.80914515239880851546838231624, −7.58209366769652586999848957206, −6.19450505012574488402066678964, −5.32545282084031423616502472636, −3.2943499622328760015142376625, −2.40911414521744057912588289218, 2.12652156444022943100733545988, 3.85713118016557444500966954739, 5.159131292285375575767094879007, 6.35120844369665879949068253931, 7.687448040669914946730215133599, 9.390277993418775098253405016350, 10.761469707202499829360733178136, 12.37964110232101448973285103294, 13.17617068766516122540723753170, 14.04660600323492631798815279232, 15.61391760755120163604967593870, 16.49754115880759754421758996854, 17.45892686970901663603041570977, 19.37098276622177645600415963280, 20.44147234424034250627594532035, 21.39997954846159124661520578707, 22.34226381545322554277350952331, 23.78860670003859176553086421770, 24.14952790892503858958692781257, 25.85517968961375494435031013468, 26.07199296724238850578858047861, 28.27059379779268454287572715109, 29.0702584220541991389659341715, 29.926600502791933541361149220967, 31.30865616916645049588247738282

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