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Properties

Label 1-648-648.157-r0-0-0
Degree $1$
Conductor $648$
Sign $-0.431 - 0.902i$
Analytic cond. $3.00929$
Root an. cond. $3.00929$
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  + (0.286 − 0.957i)5-s + (0.396 − 0.918i)7-s + (−0.973 − 0.230i)11-s + (0.835 − 0.549i)13-s + (−0.939 − 0.342i)17-s + (0.939 − 0.342i)19-s + (0.396 + 0.918i)23-s + (−0.835 − 0.549i)25-s + (0.0581 − 0.998i)29-s + (−0.993 − 0.116i)31-s + (−0.766 − 0.642i)35-s + (−0.766 + 0.642i)37-s + (0.893 + 0.448i)41-s + (0.686 − 0.727i)43-s + (−0.993 + 0.116i)47-s + ⋯
L(s)  = 1  + (0.286 − 0.957i)5-s + (0.396 − 0.918i)7-s + (−0.973 − 0.230i)11-s + (0.835 − 0.549i)13-s + (−0.939 − 0.342i)17-s + (0.939 − 0.342i)19-s + (0.396 + 0.918i)23-s + (−0.835 − 0.549i)25-s + (0.0581 − 0.998i)29-s + (−0.993 − 0.116i)31-s + (−0.766 − 0.642i)35-s + (−0.766 + 0.642i)37-s + (0.893 + 0.448i)41-s + (0.686 − 0.727i)43-s + (−0.993 + 0.116i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.431 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.431 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $-0.431 - 0.902i$
Analytic conductor: \(3.00929\)
Root analytic conductor: \(3.00929\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 648,\ (0:\ ),\ -0.431 - 0.902i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6878621206 - 1.091367552i\)
\(L(\frac12)\) \(\approx\) \(0.6878621206 - 1.091367552i\)
\(L(1)\) \(\approx\) \(0.9821552129 - 0.4413526424i\)
\(L(1)\) \(\approx\) \(0.9821552129 - 0.4413526424i\)

Euler product

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

−22.93719118629834419683145316361, −22.38062664364671057986250581923, −21.39226705188442524532464476733, −20.97386161056314538195569036666, −19.809192407174829682157029017131, −18.74570682867736816358335732532, −18.19328137757536185611601268289, −17.78611690222731898813585467317, −16.32293172401475273065865581305, −15.64122245526097625800516082841, −14.78716102179923616666848692143, −14.10401700776690679355360896054, −13.1102288439480320797535043040, −12.22955047779384488785905127729, −11.03586026025266060308704475187, −10.7562561626514088814537862369, −9.46663408098212354728353452750, −8.67825613137354493159706104355, −7.641211780518720994515801883924, −6.677712027879753036828321630818, −5.8023026403198880270786388414, −4.91125111600411317987759707747, −3.56790814425993800871083992701, −2.55601341315147645341658872714, −1.7417028877093498989375108211, 0.63224927985001193751355219635, 1.69594637574095347851786787690, 3.08883310707005897840838576012, 4.24683246462142008092692610742, 5.11545352199020578812429630677, 5.90265082613209261043091172469, 7.27833379360689106453962791087, 7.9996330581976922874889590443, 8.91997216132461684605619818758, 9.8335360130116229435457749083, 10.83595708755746316785891269359, 11.502475011291340812140487745264, 12.794456921815297443379192145193, 13.46995284205791596213046005851, 13.88159455135480035255188062700, 15.42161578514150254190164550029, 15.92189509085608260774073988575, 16.87296819581070784240928810878, 17.67830662083596879961205720543, 18.24933820560242127391027235752, 19.54887670723388924225285158212, 20.36554494735952499088427607098, 20.77740767413796992659459647428, 21.604131252377642475163132079450, 22.75883176956609461693961307177

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