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Properties

Label 1-3e5-243.101-r1-0-0
Degree $1$
Conductor $243$
Sign $0.522 + 0.852i$
Analytic cond. $26.1139$
Root an. cond. $26.1139$
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  + (0.431 − 0.902i)2-s + (−0.627 − 0.778i)4-s + (0.740 − 0.672i)5-s + (−0.981 + 0.192i)7-s + (−0.973 + 0.230i)8-s + (−0.286 − 0.957i)10-s + (−0.533 + 0.845i)11-s + (−0.910 − 0.413i)13-s + (−0.249 + 0.968i)14-s + (−0.211 + 0.977i)16-s + (0.835 + 0.549i)17-s + (0.893 + 0.448i)19-s + (−0.987 − 0.154i)20-s + (0.533 + 0.845i)22-s + (−0.657 + 0.753i)23-s + ⋯
L(s)  = 1  + (0.431 − 0.902i)2-s + (−0.627 − 0.778i)4-s + (0.740 − 0.672i)5-s + (−0.981 + 0.192i)7-s + (−0.973 + 0.230i)8-s + (−0.286 − 0.957i)10-s + (−0.533 + 0.845i)11-s + (−0.910 − 0.413i)13-s + (−0.249 + 0.968i)14-s + (−0.211 + 0.977i)16-s + (0.835 + 0.549i)17-s + (0.893 + 0.448i)19-s + (−0.987 − 0.154i)20-s + (0.533 + 0.845i)22-s + (−0.657 + 0.753i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(243\)    =    \(3^{5}\)
Sign: $0.522 + 0.852i$
Analytic conductor: \(26.1139\)
Root analytic conductor: \(26.1139\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{243} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 243,\ (1:\ ),\ 0.522 + 0.852i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5316760965 + 0.2978657996i\)
\(L(\frac12)\) \(\approx\) \(0.5316760965 + 0.2978657996i\)
\(L(1)\) \(\approx\) \(0.8629522705 - 0.4194176295i\)
\(L(1)\) \(\approx\) \(0.8629522705 - 0.4194176295i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 + (0.431 - 0.902i)T \)
5 \( 1 + (0.740 - 0.672i)T \)
7 \( 1 + (-0.981 + 0.192i)T \)
11 \( 1 + (-0.533 + 0.845i)T \)
13 \( 1 + (-0.910 - 0.413i)T \)
17 \( 1 + (0.835 + 0.549i)T \)
19 \( 1 + (0.893 + 0.448i)T \)
23 \( 1 + (-0.657 + 0.753i)T \)
29 \( 1 + (-0.713 + 0.700i)T \)
31 \( 1 + (-0.875 + 0.483i)T \)
37 \( 1 + (0.396 - 0.918i)T \)
41 \( 1 + (-0.996 - 0.0774i)T \)
43 \( 1 + (-0.135 - 0.990i)T \)
47 \( 1 + (0.875 + 0.483i)T \)
53 \( 1 + (-0.173 + 0.984i)T \)
59 \( 1 + (-0.466 + 0.884i)T \)
61 \( 1 + (-0.627 + 0.778i)T \)
67 \( 1 + (0.713 + 0.700i)T \)
71 \( 1 + (0.686 - 0.727i)T \)
73 \( 1 + (-0.286 + 0.957i)T \)
79 \( 1 + (-0.565 - 0.824i)T \)
83 \( 1 + (-0.996 + 0.0774i)T \)
89 \( 1 + (0.686 + 0.727i)T \)
97 \( 1 + (-0.740 - 0.672i)T \)
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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

−25.85032062335893446810450318121, −24.8388604674853696174588346550, −24.026589634920031218367918732141, −22.93408033571820541505275729346, −22.15819040293156947249870130478, −21.62547337086772261230616238698, −20.36959670907807303225430869772, −18.8302499872716759970247362022, −18.35277087858951511673210308959, −17.00377524787941917301741489740, −16.4436680068592514392414001543, −15.38873612335780299632771750547, −14.28435737157156418003240280826, −13.65855762434288986510129148555, −12.77203213229874191637231398981, −11.52261507078736001908960435476, −9.97802432122879038595928315462, −9.378706083030386513816141482351, −7.85420787876069262276935989115, −6.91383927070960487818237380793, −6.033579707969266632531690530815, −5.11074974949493876057905478384, −3.51000297222595201056806681541, −2.647710961578347127494469909279, −0.16280029331641798837789824346, 1.4579804417019983203038183466, 2.600826349397342660523102292623, 3.8012524228082056118978324618, 5.25633430697774358335276219357, 5.76274044020542369090430645587, 7.4384211203685648797440697831, 9.05627044935295518058225923685, 9.82887634465800086341571671131, 10.4088811593943559009268711599, 12.203582832042166731322730736309, 12.52677491482060833686881100162, 13.45991720867402200024610516297, 14.4801465741607539892548244367, 15.591387977406030615168045730860, 16.78603993636555949613599767672, 17.85607193002235751190355766542, 18.724029886532157044042248244546, 19.95310866330799901506502146529, 20.35132170400001637241500253765, 21.564986095841847066243075699696, 22.13874621685681500201073237930, 23.14088098291317104485921725735, 24.04402030439743791647282119058, 25.16963738105547300519418413742, 25.937493188777574470504328409310

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