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Properties

Label 1-3e4-81.34-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  + (−0.686 − 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)  = 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

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

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: $\chi_{81} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 81,\ (0:\ ),\ 0.996 - 0.0774i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6657570992 + 0.02583440739i\)
\(L(\frac12)\) \(\approx\) \(0.6657570992 + 0.02583440739i\)
\(L(1)\) \(\approx\) \(0.7316518116 - 0.08858051385i\)
\(L(1)\) \(\approx\) \(0.7316518116 - 0.08858051385i\)

Euler product

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

−31.0409248027540366922801992774, −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