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Properties

Label 1-648-648.619-r1-0-0
Degree $1$
Conductor $648$
Sign $0.713 - 0.700i$
Analytic cond. $69.6372$
Root an. cond. $69.6372$
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  + (0.0581 − 0.998i)5-s + (−0.973 + 0.230i)7-s + (−0.835 + 0.549i)11-s + (0.993 − 0.116i)13-s + (0.766 + 0.642i)17-s + (0.766 − 0.642i)19-s + (−0.973 − 0.230i)23-s + (−0.993 − 0.116i)25-s + (−0.597 + 0.802i)29-s + (0.286 + 0.957i)31-s + (0.173 + 0.984i)35-s + (−0.173 + 0.984i)37-s + (0.396 − 0.918i)41-s + (0.893 + 0.448i)43-s + (0.286 − 0.957i)47-s + ⋯
L(s)  = 1  + (0.0581 − 0.998i)5-s + (−0.973 + 0.230i)7-s + (−0.835 + 0.549i)11-s + (0.993 − 0.116i)13-s + (0.766 + 0.642i)17-s + (0.766 − 0.642i)19-s + (−0.973 − 0.230i)23-s + (−0.993 − 0.116i)25-s + (−0.597 + 0.802i)29-s + (0.286 + 0.957i)31-s + (0.173 + 0.984i)35-s + (−0.173 + 0.984i)37-s + (0.396 − 0.918i)41-s + (0.893 + 0.448i)43-s + (0.286 − 0.957i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.713 - 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.713 - 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.713 - 0.700i$
Analytic conductor: \(69.6372\)
Root analytic conductor: \(69.6372\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (619, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 648,\ (1:\ ),\ 0.713 - 0.700i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.481016769 - 0.6050619309i\)
\(L(\frac12)\) \(\approx\) \(1.481016769 - 0.6050619309i\)
\(L(1)\) \(\approx\) \(0.9773498070 - 0.1377653054i\)
\(L(1)\) \(\approx\) \(0.9773498070 - 0.1377653054i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (0.0581 - 0.998i)T \)
7 \( 1 + (-0.973 + 0.230i)T \)
11 \( 1 + (-0.835 + 0.549i)T \)
13 \( 1 + (0.993 - 0.116i)T \)
17 \( 1 + (0.766 + 0.642i)T \)
19 \( 1 + (0.766 - 0.642i)T \)
23 \( 1 + (-0.973 - 0.230i)T \)
29 \( 1 + (-0.597 + 0.802i)T \)
31 \( 1 + (0.286 + 0.957i)T \)
37 \( 1 + (-0.173 + 0.984i)T \)
41 \( 1 + (0.396 - 0.918i)T \)
43 \( 1 + (0.893 + 0.448i)T \)
47 \( 1 + (0.286 - 0.957i)T \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 + (-0.835 - 0.549i)T \)
61 \( 1 + (0.686 + 0.727i)T \)
67 \( 1 + (0.597 + 0.802i)T \)
71 \( 1 + (0.939 - 0.342i)T \)
73 \( 1 + (-0.939 - 0.342i)T \)
79 \( 1 + (-0.396 - 0.918i)T \)
83 \( 1 + (0.396 + 0.918i)T \)
89 \( 1 + (-0.939 - 0.342i)T \)
97 \( 1 + (-0.0581 - 0.998i)T \)
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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

−22.91825667133844901566612256191, −22.06299275558503156734218042527, −21.138049220589297255065381746534, −20.386728250168435389211581823576, −19.26800006621038788774411985112, −18.610236691020570466051761841943, −18.15802754350601534061370084941, −16.8807141842541457274699214698, −15.95800275566535565846593742077, −15.56710926871757687753197405705, −14.16908378380712964612412553338, −13.7617326120199641384379659983, −12.81733536437363428412711240972, −11.6937735167347245859531683141, −10.88475567559283703579564014323, −10.039326471328245433406965484034, −9.38092939819501185668869273147, −7.93150996309568150455319441765, −7.36796551259312304453704351725, −6.06270979391606012821142038037, −5.775482267479429347735378724360, −3.97409256770122591622987759056, −3.27303211798957830444372003908, −2.36883146706648780358307105725, −0.7396026515268726198445813226, 0.55633736814925247789356864837, 1.74026649913051613820474825449, 3.05947028471119406008696296777, 4.01976129303631830588986062869, 5.23186561829307348720416974014, 5.86593068325723356202323393456, 7.0276847891665704571978591426, 8.11970227668634651070888118366, 8.87972977857141758240896698720, 9.80397944074577042549281320178, 10.520883693401301582873729776540, 11.8410812422847895609505446129, 12.62239000119417708035078275360, 13.165404062441816627977390984461, 14.04624883362669896249906663578, 15.410288987734439367758517663065, 15.96316519937532274738383601242, 16.58123217131740810907645552438, 17.67153549846517938068485401612, 18.41095899034630989582153228001, 19.376988538025648425477522408850, 20.20889821109620846883563630044, 20.80826494189073729277030327177, 21.70044002213273177214547253881, 22.61097510149486056502336112080

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