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Properties

Label 1-648-648.259-r1-0-0
Degree $1$
Conductor $648$
Sign $-0.431 + 0.902i$
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  + (0.286 + 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

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

Invariants

Degree: \(1\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $-0.431 + 0.902i$
Analytic conductor: \(69.6372\)
Root analytic conductor: \(69.6372\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 648,\ (1:\ ),\ -0.431 + 0.902i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7621498563 + 1.209233069i\)
\(L(\frac12)\) \(\approx\) \(0.7621498563 + 1.209233069i\)
\(L(1)\) \(\approx\) \(1.017498919 + 0.2236461827i\)
\(L(1)\) \(\approx\) \(1.017498919 + 0.2236461827i\)

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

−22.44779947701521851095112791163, −21.52500658323388392696927666273, −20.78297892123460419685848729058, −19.989254104835883032639253481299, −19.199157241895178953055104872, −18.23390291823834424208956071319, −17.41808731541370102482185823357, −16.63059899669978850242389834350, −15.7434417158579272253694101776, −15.11345479149016384038587407571, −13.91437327598681640011211560831, −13.107564491045277430655590332180, −12.33233365700292410964432185708, −11.67079436768397798459781415078, −10.44318434540849073731837981471, −9.43688551746433365685204951977, −8.750465173651793720187957709345, −8.12128578732860430749227829926, −6.43194883595481528403516604503, −6.07750624062004293107397394657, −4.815017026854591376881587823, −4.012865580872340383657103141579, −2.62500832644582348969897437103, −1.61082671260233827960280948725, −0.34657975638206369589389132078, 1.20553337267391962329583770072, 2.35024899754525955473801856050, 3.69138012710810463761105995754, 4.11262904364133675231874416189, 5.7787336234017747931855624917, 6.70998562445754784400696491691, 7.03747726717200282894690088483, 8.48907144430272811732538121121, 9.330028962975855768338439294930, 10.40309542291380197182135401023, 10.96266879127740339047884561537, 11.81625684452831919950342445264, 13.15501195382535740338906080665, 13.77468375930990642959510906185, 14.459964190101465813556935061830, 15.45334629716078791141978797325, 16.323396359256163355649784209579, 17.34161462260391966580518502996, 17.79108340115548777960172846297, 19.09360385120523199218093888535, 19.42007680020516371698347454429, 20.42372212742686801449234009951, 21.493454589089296847138544548890, 22.07819164814922355457744280608, 22.94429458659416989860275321083

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