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Properties

Label 1-648-648.499-r1-0-0
Degree $1$
Conductor $648$
Sign $0.249 - 0.968i$
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  + (0.835 − 0.549i)5-s + (0.686 − 0.727i)7-s + (0.893 − 0.448i)11-s + (−0.396 − 0.918i)13-s + (0.766 − 0.642i)17-s + (0.766 + 0.642i)19-s + (0.686 + 0.727i)23-s + (0.396 − 0.918i)25-s + (0.993 − 0.116i)29-s + (−0.973 + 0.230i)31-s + (0.173 − 0.984i)35-s + (−0.173 − 0.984i)37-s + (0.597 − 0.802i)41-s + (−0.0581 + 0.998i)43-s + (−0.973 − 0.230i)47-s + ⋯
L(s)  = 1  + (0.835 − 0.549i)5-s + (0.686 − 0.727i)7-s + (0.893 − 0.448i)11-s + (−0.396 − 0.918i)13-s + (0.766 − 0.642i)17-s + (0.766 + 0.642i)19-s + (0.686 + 0.727i)23-s + (0.396 − 0.918i)25-s + (0.993 − 0.116i)29-s + (−0.973 + 0.230i)31-s + (0.173 − 0.984i)35-s + (−0.173 − 0.984i)37-s + (0.597 − 0.802i)41-s + (−0.0581 + 0.998i)43-s + (−0.973 − 0.230i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.249 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.249 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.249 - 0.968i$
Analytic conductor: \(69.6372\)
Root analytic conductor: \(69.6372\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 648,\ (1:\ ),\ 0.249 - 0.968i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.487956723 - 1.928311969i\)
\(L(\frac12)\) \(\approx\) \(2.487956723 - 1.928311969i\)
\(L(1)\) \(\approx\) \(1.460649760 - 0.4543153647i\)
\(L(1)\) \(\approx\) \(1.460649760 - 0.4543153647i\)

Euler product

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

−22.60923586383110381508234401526, −21.928216068353283185279209830035, −21.38392598948082191647427740278, −20.53874206426902897484261438432, −19.41046408051193816945751204392, −18.6797016397520055167169617805, −17.86851737383920304824326483823, −17.20726507637225540681423423626, −16.36307440163976563515062072466, −15.006643399208597649224535655757, −14.59124597468869671732531939321, −13.87818328249974873184607570695, −12.71551832915708808540284166729, −11.83024419972075456343407712649, −11.11605618923400522699133902312, −9.9724871211804163757668299956, −9.296404292475537797226448483428, −8.42067492112089932775347857923, −7.13176140932114256073487552699, −6.45554928146853770339155541660, −5.39926166307858911279816611717, −4.55644490610125830629403954199, −3.183677362665396747705870077380, −2.131858940404537781525403349980, −1.31871398314775154897137210916, 0.82828540224492994245311994126, 1.44044991000120246450284538009, 2.89469006125577163022150004469, 4.00406269467908900224394047871, 5.18139491333875599529924345761, 5.713353785142974768745887914562, 7.04484285419304110508333272254, 7.85860201584542053649908062129, 8.89784137724040947510254694651, 9.74635538196822306236380849162, 10.52824553825000355616478514977, 11.56834432679822339201069992974, 12.42773246075003919597224936466, 13.41753185318306538343796241902, 14.154939401842190486682923620634, 14.71189062885451613427018525523, 16.149709498789382189144093211426, 16.73075335380478908543489278441, 17.61363759100022384037485236051, 18.05317761364706422694822377203, 19.40194329669926106892323864091, 20.11531758655007991079998517101, 20.915530131476009804274843938581, 21.49557675641228854608975279491, 22.53214873213148450456267808613

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