Properties

Degree 1
Conductor $ 3^{4} $
Sign $0.996 - 0.0774i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.686 − 0.727i)2-s + (−0.0581 + 0.998i)4-s + (−0.993 + 0.116i)5-s + (0.893 + 0.448i)7-s + (0.766 − 0.642i)8-s + (0.766 + 0.642i)10-s + (0.396 + 0.918i)11-s + (0.973 + 0.230i)13-s + (−0.286 − 0.957i)14-s + (−0.993 − 0.116i)16-s + (0.173 − 0.984i)17-s + (0.173 + 0.984i)19-s + (−0.0581 − 0.998i)20-s + (0.396 − 0.918i)22-s + (0.893 − 0.448i)23-s + ⋯
L(s,χ)  = 1  + (−0.686 − 0.727i)2-s + (−0.0581 + 0.998i)4-s + (−0.993 + 0.116i)5-s + (0.893 + 0.448i)7-s + (0.766 − 0.642i)8-s + (0.766 + 0.642i)10-s + (0.396 + 0.918i)11-s + (0.973 + 0.230i)13-s + (−0.286 − 0.957i)14-s + (−0.993 − 0.116i)16-s + (0.173 − 0.984i)17-s + (0.173 + 0.984i)19-s + (−0.0581 − 0.998i)20-s + (0.396 − 0.918i)22-s + (0.893 − 0.448i)23-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(81\)    =    \(3^{4}\)
\( \varepsilon \)  =  $0.996 - 0.0774i$
motivic weight  =  \(0\)
character  :  $\chi_{81} (34, \cdot )$
Sato-Tate  :  $\mu(27)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 81,\ (0:\ ),\ 0.996 - 0.0774i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.6657570992 + 0.02583440739i$
$L(\frac12,\chi)$  $\approx$  $0.6657570992 + 0.02583440739i$
$L(\chi,1)$  $\approx$  0.7316518116 - 0.08858051385i
$L(1,\chi)$  $\approx$  0.7316518116 - 0.08858051385i

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−31.0409248027540366922801992774, −30.02105735442885264076762837340, −28.45714564150378403951349136805, −27.57640555911783753004297969782, −26.88568301635513074492311657147, −25.84705363335257416275338787898, −24.38892321329929140873132622063, −23.846196706497307170941505247769, −22.844911624483944643522945780397, −21.08865984991571414035380250033, −19.80600856417433526358765978461, −19.003161444958792890299674516037, −17.73928247019524183672981001724, −16.71216163733850617278973185780, −15.641384720519435877415059283580, −14.68205414654886099322585179580, −13.41988975101954881748368576901, −11.40526983907931459976114928764, −10.759906424188829758194215992185, −8.84602284422507120297256237636, −8.104699579140437505822357144064, −6.915767366585370571515089204729, −5.35878487645804882569785278096, −3.847060916607352497141398823634, −1.147169855495278158729360665145, 1.60374649004089874975783527944, 3.38749118547016160069622309004, 4.71767415251699233223905419943, 7.09001228378691396133564310085, 8.16593145860477571543289318338, 9.23111358826437091777313114294, 10.828268129697150915937694861599, 11.676755514627017334707635604655, 12.58194011160752478817783774810, 14.3620215311603872268276264511, 15.67154612038522290167264497248, 16.84991122498506511608278327624, 18.248700203778217888341303182074, 18.80359864615634513165452372365, 20.28811457527556162086159341728, 20.79745093555547472737835925237, 22.2974934239045674360248486486, 23.2787790132367730871405361935, 24.78060966096400860816240471069, 25.84186376066469855134319016410, 27.248152933820678210103718597724, 27.6048923857041673191060129046, 28.65931383686744381725539560291, 30.00304900693114064296270440019, 31.05240217386687407325188163341

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