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Properties

Label 1-3e4-81.79-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  + (−0.993 − 0.116i)2-s + (0.973 + 0.230i)4-s + (0.893 − 0.448i)5-s + (−0.286 + 0.957i)7-s + (−0.939 − 0.342i)8-s + (−0.939 + 0.342i)10-s + (−0.0581 − 0.998i)11-s + (0.597 + 0.802i)13-s + (0.396 − 0.918i)14-s + (0.893 + 0.448i)16-s + (0.766 + 0.642i)17-s + (0.766 − 0.642i)19-s + (0.973 − 0.230i)20-s + (−0.0581 + 0.998i)22-s + (−0.286 − 0.957i)23-s + ⋯
L(s)  = 1  + (−0.993 − 0.116i)2-s + (0.973 + 0.230i)4-s + (0.893 − 0.448i)5-s + (−0.286 + 0.957i)7-s + (−0.939 − 0.342i)8-s + (−0.939 + 0.342i)10-s + (−0.0581 − 0.998i)11-s + (0.597 + 0.802i)13-s + (0.396 − 0.918i)14-s + (0.893 + 0.448i)16-s + (0.766 + 0.642i)17-s + (0.766 − 0.642i)19-s + (0.973 − 0.230i)20-s + (−0.0581 + 0.998i)22-s + (−0.286 − 0.957i)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: $\chi_{81} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 81,\ (0:\ ),\ 0.996 - 0.0774i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7338024755 + 0.02847487788i\)
\(L(\frac12)\) \(\approx\) \(0.7338024755 + 0.02847487788i\)
\(L(1)\) \(\approx\) \(0.7903779748 + 0.02932987410i\)
\(L(1)\) \(\approx\) \(0.7903779748 + 0.02932987410i\)

Euler product

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

−30.55701779624211275085959815417, −29.72646037262166390415007887830, −28.93848123062537913171495593609, −27.72925707370024731637561482914, −26.71221259249183400693154364271, −25.614554972445407143602838352839, −25.19255019685459071374540164582, −23.56116326309607292273817155893, −22.484859165468284118324012821695, −20.8037495500588811347322757498, −20.2417098890790849125606735606, −18.76473126175363901146504437265, −17.818166412991566444553231833584, −17.01482130041922819895879682815, −15.773593194345885757929914689957, −14.4673870399616747121479667503, −13.206340126145945756240228954507, −11.53681914402553463982594267839, −10.05368943902825942482520394057, −9.8182859348706313353223055664, −7.89194634722407838512945088082, −6.89628939201845493675705672781, −5.586847104583670306328294024113, −3.21123333482134676854408223536, −1.50120548673492442729048298941, 1.54887701770977937442511451377, 3.05343002450603353015584344131, 5.559741354438408090208429253854, 6.56941601463254818571318780825, 8.48773235132545478514735362599, 9.11425568973302878511564615568, 10.3679088021711185390025732372, 11.69868418623004272543066613664, 12.87894472652382113220280717974, 14.38628231788394195260542529329, 16.032036522730955179680261681523, 16.641267081011651344186289704545, 18.06608552547573329341162282728, 18.73791081306528977834774111779, 19.99190041316430376601914446415, 21.34316873940910365808346058164, 21.77488724742358337747076942767, 23.91078449138515202423166479870, 24.85171023255120972072118587464, 25.7179928175184091725595183078, 26.64269929948755088778907694473, 28.11961842634769876439692588400, 28.63448449628008300666598789848, 29.55843493575035940891115885841, 30.78110116871050085088573258099

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