↑

Properties

Label 1-3e4-81.22-r0-0-0
Degree $1$
Conductor $81$
Sign $0.0968 + 0.995i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0581 + 0.998i)2-s + (−0.993 − 0.116i)4-s + (0.973 − 0.230i)5-s + (0.597 + 0.802i)7-s + (0.173 − 0.984i)8-s + (0.173 + 0.984i)10-s + (−0.686 + 0.727i)11-s + (0.893 + 0.448i)13-s + (−0.835 + 0.549i)14-s + (0.973 + 0.230i)16-s + (−0.939 − 0.342i)17-s + (−0.939 + 0.342i)19-s + (−0.993 + 0.116i)20-s + (−0.686 − 0.727i)22-s + (0.597 − 0.802i)23-s + ⋯
L(s)  = 1  + (−0.0581 + 0.998i)2-s + (−0.993 − 0.116i)4-s + (0.973 − 0.230i)5-s + (0.597 + 0.802i)7-s + (0.173 − 0.984i)8-s + (0.173 + 0.984i)10-s + (−0.686 + 0.727i)11-s + (0.893 + 0.448i)13-s + (−0.835 + 0.549i)14-s + (0.973 + 0.230i)16-s + (−0.939 − 0.342i)17-s + (−0.939 + 0.342i)19-s + (−0.993 + 0.116i)20-s + (−0.686 − 0.727i)22-s + (0.597 − 0.802i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0968 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0968 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(81\)    =    \(3^{4}\)
Sign: $0.0968 + 0.995i$
Analytic conductor: \(0.376162\)
Root analytic conductor: \(0.376162\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{81} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 81,\ (0:\ ),\ 0.0968 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7404552842 + 0.6719273433i\)
\(L(\frac12)\) \(\approx\) \(0.7404552842 + 0.6719273433i\)
\(L(1)\) \(\approx\) \(0.9109849642 + 0.5334985951i\)
\(L(1)\) \(\approx\) \(0.9109849642 + 0.5334985951i\)

Euler product

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

Imaginary part of the first few zeros on the critical line

−30.39828911869058810625361852054, −29.74709619129973350041102187197, −28.83739070478102770425556552827, −27.72508206793947145378391167277, −26.58668505864200477749867140844, −25.77291462574095172992217237699, −24.09616315024363280596614451370, −23.06569252977329530745503425278, −21.752986510152455334600781344107, −21.05209738322340936715023610149, −20.09214673353839363884491756576, −18.69715229093568804550706553446, −17.79231805840609575078113863472, −16.89227762950962207962065836504, −14.91861148245749097408555779542, −13.459654905027872434755464453932, −13.217623195794901364045866712743, −11.07127927059037263451464529943, −10.70314371361620846084698749085, −9.254851094640615781039281659083, −8.002902726759443833341305732842, −6.0208524936554592081327124008, −4.56299343697567634235595656035, −2.9755007393240368736737084078, −1.4479441812529719786381235513, 2.05450028644898398116301178421, 4.541165499558567760715165613372, 5.61853700754692715941359835036, 6.75949259448805605852981760594, 8.41001201841857183147924176460, 9.212298094002598992335445944088, 10.646258577308296930929432481651, 12.58118065513609780647048507000, 13.581362917040254131951456922802, 14.750039837009186405351653188166, 15.73843611353678655587488796359, 17.030283772829479434354187269, 17.99688902292953567794904107235, 18.70240842034910860705532497120, 20.72490049914285390275871871251, 21.60155832017538735677225405010, 22.80750285417904185149793608329, 23.982397026812508466688358373335, 24.99081748843174119827314310643, 25.647617675307911703010514858294, 26.75957638663801881788245752485, 28.15176078437410164478693117810, 28.70977781342254806902253798402, 30.51929028243888528111954612517, 31.40584372602693181979932031726

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