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Properties

Label 1-633-633.176-r1-0-0
Degree $1$
Conductor $633$
Sign $0.186 - 0.982i$
Analytic cond. $68.0252$
Root an. cond. $68.0252$
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  + (−0.251 + 0.967i)2-s + (−0.873 − 0.486i)4-s + (0.550 − 0.834i)5-s + (0.712 − 0.701i)7-s + (0.691 − 0.722i)8-s + (0.669 + 0.743i)10-s + (−0.936 − 0.351i)11-s + (0.473 − 0.880i)13-s + (0.5 + 0.866i)14-s + (0.525 + 0.850i)16-s + (0.337 − 0.941i)17-s + (−0.104 − 0.994i)19-s + (−0.887 + 0.460i)20-s + (0.575 − 0.817i)22-s + (0.809 + 0.587i)23-s + ⋯
L(s)  = 1  + (−0.251 + 0.967i)2-s + (−0.873 − 0.486i)4-s + (0.550 − 0.834i)5-s + (0.712 − 0.701i)7-s + (0.691 − 0.722i)8-s + (0.669 + 0.743i)10-s + (−0.936 − 0.351i)11-s + (0.473 − 0.880i)13-s + (0.5 + 0.866i)14-s + (0.525 + 0.850i)16-s + (0.337 − 0.941i)17-s + (−0.104 − 0.994i)19-s + (−0.887 + 0.460i)20-s + (0.575 − 0.817i)22-s + (0.809 + 0.587i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 633 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.186 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 633 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.186 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(633\)    =    \(3 \cdot 211\)
Sign: $0.186 - 0.982i$
Analytic conductor: \(68.0252\)
Root analytic conductor: \(68.0252\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{633} (176, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 633,\ (1:\ ),\ 0.186 - 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.391536407 - 1.152340950i\)
\(L(\frac12)\) \(\approx\) \(1.391536407 - 1.152340950i\)
\(L(1)\) \(\approx\) \(1.051316781 - 0.05862158094i\)
\(L(1)\) \(\approx\) \(1.051316781 - 0.05862158094i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
211 \( 1 \)
good2 \( 1 + (-0.251 + 0.967i)T \)
5 \( 1 + (0.550 - 0.834i)T \)
7 \( 1 + (0.712 - 0.701i)T \)
11 \( 1 + (-0.936 - 0.351i)T \)
13 \( 1 + (0.473 - 0.880i)T \)
17 \( 1 + (0.337 - 0.941i)T \)
19 \( 1 + (-0.104 - 0.994i)T \)
23 \( 1 + (0.809 + 0.587i)T \)
29 \( 1 + (0.999 - 0.0299i)T \)
31 \( 1 + (0.826 - 0.563i)T \)
37 \( 1 + (-0.163 + 0.986i)T \)
41 \( 1 + (0.925 - 0.379i)T \)
43 \( 1 + (0.0747 + 0.997i)T \)
47 \( 1 + (-0.992 - 0.119i)T \)
53 \( 1 + (0.873 - 0.486i)T \)
59 \( 1 + (-0.791 - 0.611i)T \)
61 \( 1 + (-0.978 + 0.207i)T \)
67 \( 1 + (0.623 + 0.781i)T \)
71 \( 1 + (-0.309 + 0.951i)T \)
73 \( 1 + (-0.733 + 0.680i)T \)
79 \( 1 + (-0.691 - 0.722i)T \)
83 \( 1 + (0.978 + 0.207i)T \)
89 \( 1 + (0.0448 - 0.998i)T \)
97 \( 1 + (-0.995 - 0.0896i)T \)
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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

−22.92478195569128786788589970026, −21.66227324856630233310805281820, −21.26729126286168791868258796763, −20.80905439726309007340643437627, −19.40475606169450402556476578354, −18.76723017769302786752154067839, −18.152298472515430300551613659282, −17.53122051760632057802400552170, −16.49252933989690019161959126697, −15.19909011177594496522775652477, −14.3896346039139514967718387923, −13.698784309487731539306774215257, −12.60203182482805338967889063767, −11.93530680909214758262988864023, −10.759517623864728341983196746651, −10.50822657021225686287611618815, −9.37344863876023411944069663506, −8.48362677503936892031900818879, −7.656650226141747304070757747, −6.31701774814126463786465337742, −5.30066464894077670463903742350, −4.26760336823804861802954071859, −3.051287489155473437335129576983, −2.20405179834719226475767014640, −1.40810923981066764313256968580, 0.532018794782938447895216840909, 1.190024163233035498567138494025, 2.91602178363660899586439627480, 4.53047238747057996208401933064, 5.06305058078914932809210079228, 5.88978535548956652952258008195, 7.07621922127980901482445246227, 8.00198141522092333523675844064, 8.55743144800450293635130182421, 9.64468516736100406433393988315, 10.39979789168544854223013231942, 11.41599533504632791858242150515, 12.94544574739407691578475597119, 13.43903152637372170677762181885, 14.09588845681569185242408659260, 15.27491144698501054295127637255, 15.95428915004879994973082399404, 16.76057460857854101456724011625, 17.672702119731336835542356387970, 17.92558678859493469059343804059, 19.11278510661155452163176067105, 20.1650788326191883316525248800, 20.92486133932708799112440682709, 21.66129276275517265158715563691, 23.01721557803675229947951861026

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