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Properties

Label 1-3e4-81.40-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: $\chi_{81} (40, \cdot )$
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

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

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