↑

Properties

Label 1-3e4-81.7-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

Related objects

Downloads

Learn more

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

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

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: $\chi_{81} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 81,\ (0:\ ),\ 0.996 - 0.0774i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8092290392 - 0.03140177205i\)
\(L(\frac12)\) \(\approx\) \(0.8092290392 - 0.03140177205i\)
\(L(1)\) \(\approx\) \(0.8773440488 + 0.08884031705i\)
\(L(1)\) \(\approx\) \(0.8773440488 + 0.08884031705i\)

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 \( 1 + (0.396 + 0.918i)T \)
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

−30.53842512459090959200104893415, −30.191967818154680515741212602411, −28.79856507380144963540202116643, −28.05796485020104373435301004647, −26.89280018558609185288038551284, −25.86754009684195243827409962205, −24.88839150889167163967813910781, −22.99753225801243068980915971231, −22.112664347146054370569660326, −21.48674392449048532120534535836, −19.995985478743479957194710260922, −19.08341037448854537482683699666, −17.98876273987377204890058233365, −17.290301217723035959023677286295, −15.28402702147323462844370026414, −14.291485997383692573309265417222, −12.78339249853589157071294471072, −11.87862721587121773161036985308, −10.529928483997389508686048446987, −9.638903117979400331412141431235, −8.326195054374422588748065653490, −6.663063151445585778653304100297, −4.94333877261015674566801130910, −3.132383613037413262265020775256, −2.06180661431055941849415281393, 1.139416652998080874615249687570, 4.05809342870104658129554276851, 5.292706427554111764736076966614, 6.65491234550027427913368596138, 7.921951325716767974928534484661, 9.184851371335002180476225279270, 10.11353928894516492251118274436, 11.98199200322471993069434414266, 13.69167744648722752911797158305, 14.04826219042733510302370330087, 15.8548459749244012315851501846, 16.79523166214676396608023398097, 17.33735649570245633996253628230, 18.900212769186880064244713239326, 19.963765481737852957232544461933, 21.30336701141595479653368419538, 22.65647461214755379003043909675, 23.80559751476771213276425001703, 24.56739852851435665869614233789, 25.50783374442462849393203223133, 26.83538785597893435120762460234, 27.4052686082827740144392456081, 28.84461242731678947338953920326, 29.6688938294382782065099303957, 31.48951636417957451575027664687

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