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Properties

Label 1-207-207.20-r0-0-0
Degree $1$
Conductor $207$
Sign $0.999 + 0.0180i$
Analytic cond. $0.961303$
Root an. cond. $0.961303$
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  + (−0.723 − 0.690i)2-s + (0.0475 + 0.998i)4-s + (0.928 + 0.371i)5-s + (0.995 − 0.0950i)7-s + (0.654 − 0.755i)8-s + (−0.415 − 0.909i)10-s + (0.235 + 0.971i)11-s + (−0.995 − 0.0950i)13-s + (−0.786 − 0.618i)14-s + (−0.995 + 0.0950i)16-s + (0.841 + 0.540i)17-s + (−0.841 + 0.540i)19-s + (−0.327 + 0.945i)20-s + (0.5 − 0.866i)22-s + (0.723 + 0.690i)25-s + (0.654 + 0.755i)26-s + ⋯
L(s)  = 1  + (−0.723 − 0.690i)2-s + (0.0475 + 0.998i)4-s + (0.928 + 0.371i)5-s + (0.995 − 0.0950i)7-s + (0.654 − 0.755i)8-s + (−0.415 − 0.909i)10-s + (0.235 + 0.971i)11-s + (−0.995 − 0.0950i)13-s + (−0.786 − 0.618i)14-s + (−0.995 + 0.0950i)16-s + (0.841 + 0.540i)17-s + (−0.841 + 0.540i)19-s + (−0.327 + 0.945i)20-s + (0.5 − 0.866i)22-s + (0.723 + 0.690i)25-s + (0.654 + 0.755i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0180i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0180i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(207\)    =    \(3^{2} \cdot 23\)
Sign: $0.999 + 0.0180i$
Analytic conductor: \(0.961303\)
Root analytic conductor: \(0.961303\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{207} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 207,\ (0:\ ),\ 0.999 + 0.0180i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.028640207 + 0.009272232572i\)
\(L(\frac12)\) \(\approx\) \(1.028640207 + 0.009272232572i\)
\(L(1)\) \(\approx\) \(0.9342475198 - 0.09337488417i\)
\(L(1)\) \(\approx\) \(0.9342475198 - 0.09337488417i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.723 - 0.690i)T \)
5 \( 1 + (0.928 + 0.371i)T \)
7 \( 1 + (0.995 - 0.0950i)T \)
11 \( 1 + (0.235 + 0.971i)T \)
13 \( 1 + (-0.995 - 0.0950i)T \)
17 \( 1 + (0.841 + 0.540i)T \)
19 \( 1 + (-0.841 + 0.540i)T \)
29 \( 1 + (-0.0475 + 0.998i)T \)
31 \( 1 + (-0.327 - 0.945i)T \)
37 \( 1 + (0.142 - 0.989i)T \)
41 \( 1 + (-0.928 - 0.371i)T \)
43 \( 1 + (0.327 - 0.945i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (0.415 - 0.909i)T \)
59 \( 1 + (0.995 + 0.0950i)T \)
61 \( 1 + (-0.981 - 0.189i)T \)
67 \( 1 + (-0.235 + 0.971i)T \)
71 \( 1 + (0.959 - 0.281i)T \)
73 \( 1 + (0.841 - 0.540i)T \)
79 \( 1 + (-0.580 + 0.814i)T \)
83 \( 1 + (0.928 - 0.371i)T \)
89 \( 1 + (-0.654 - 0.755i)T \)
97 \( 1 + (0.786 - 0.618i)T \)
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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

−26.8607147018196348178437306965, −25.738878209899973685698030137702, −24.816382717640271823604211524501, −24.34336894782463673592019370579, −23.382811891202636653747900309108, −21.89885709320734473287006286449, −21.13189781448916557829371474904, −20.00868071274412571108893784683, −18.91491097189048194922315388081, −18.01892259621167943101295880208, −17.085592574814967946413818249592, −16.622773000653429059107696071647, −15.187077301423914127467413665935, −14.30117288509215072978214033995, −13.567724230158818944130981583544, −11.94603108509493041157223506044, −10.78975629902410418417886278542, −9.764448816580608435304053071117, −8.81051135598706639285410523050, −7.93707792059500225610012110059, −6.65189993725229070778884496587, −5.53325544544915162975782809314, −4.74526076198692715784778696324, −2.41196584223809814175550812671, −1.13728344964600602475842957095, 1.6128181593483219860456673159, 2.358367835633717151212828896921, 3.99176193733188918861884913734, 5.32830473824821306915091409790, 6.95041582098599765465611560132, 7.86767547495845827218476873301, 9.12012216059579894293899992752, 10.09850280092454954225024696960, 10.76151766545992986566730690322, 12.05753549850504610302808721506, 12.8434894360225540034423582284, 14.253274775543111572220234758475, 14.97511719740084734448572016719, 16.81892127566311648415507925130, 17.32988453638126987539722468204, 18.13380495415436695584758289275, 19.06390484530474047343725266151, 20.230931593665282179343320395365, 21.01909016875880269233078961988, 21.766929272167007983775897493602, 22.68616640739207609606465386881, 24.108211560065475118117673107950, 25.31461534661378154548750569789, 25.757588270185665842808489268343, 26.99454036501517346377898585692

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