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Properties

Label 1-12e3-1728.763-r1-0-0
Degree $1$
Conductor $1728$
Sign $0.503 + 0.864i$
Analytic cond. $185.699$
Root an. cond. $185.699$
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  + (−0.999 − 0.0436i)5-s + (−0.573 + 0.819i)7-s + (−0.675 + 0.737i)11-s + (0.953 + 0.300i)13-s + (0.866 + 0.5i)17-s + (−0.793 − 0.608i)19-s + (0.573 + 0.819i)23-s + (0.996 + 0.0871i)25-s + (−0.887 − 0.461i)29-s + (0.173 − 0.984i)31-s + (0.608 − 0.793i)35-s + (0.793 − 0.608i)37-s + (0.996 − 0.0871i)41-s + (0.675 − 0.737i)43-s + (−0.984 + 0.173i)47-s + ⋯
L(s)  = 1  + (−0.999 − 0.0436i)5-s + (−0.573 + 0.819i)7-s + (−0.675 + 0.737i)11-s + (0.953 + 0.300i)13-s + (0.866 + 0.5i)17-s + (−0.793 − 0.608i)19-s + (0.573 + 0.819i)23-s + (0.996 + 0.0871i)25-s + (−0.887 − 0.461i)29-s + (0.173 − 0.984i)31-s + (0.608 − 0.793i)35-s + (0.793 − 0.608i)37-s + (0.996 − 0.0871i)41-s + (0.675 − 0.737i)43-s + (−0.984 + 0.173i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.503 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.503 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.503 + 0.864i$
Analytic conductor: \(185.699\)
Root analytic conductor: \(185.699\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (763, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1728,\ (1:\ ),\ 0.503 + 0.864i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.182648988 + 0.6799388925i\)
\(L(\frac12)\) \(\approx\) \(1.182648988 + 0.6799388925i\)
\(L(1)\) \(\approx\) \(0.8395526155 + 0.1431592877i\)
\(L(1)\) \(\approx\) \(0.8395526155 + 0.1431592877i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-0.999 - 0.0436i)T \)
7 \( 1 + (-0.573 + 0.819i)T \)
11 \( 1 + (-0.675 + 0.737i)T \)
13 \( 1 + (0.953 + 0.300i)T \)
17 \( 1 + (0.866 + 0.5i)T \)
19 \( 1 + (-0.793 - 0.608i)T \)
23 \( 1 + (0.573 + 0.819i)T \)
29 \( 1 + (-0.887 - 0.461i)T \)
31 \( 1 + (0.173 - 0.984i)T \)
37 \( 1 + (0.793 - 0.608i)T \)
41 \( 1 + (0.996 - 0.0871i)T \)
43 \( 1 + (0.675 - 0.737i)T \)
47 \( 1 + (-0.984 + 0.173i)T \)
53 \( 1 + (-0.382 + 0.923i)T \)
59 \( 1 + (0.999 + 0.0436i)T \)
61 \( 1 + (-0.843 + 0.537i)T \)
67 \( 1 + (0.300 - 0.953i)T \)
71 \( 1 + (-0.965 + 0.258i)T \)
73 \( 1 + (0.965 + 0.258i)T \)
79 \( 1 + (-0.642 - 0.766i)T \)
83 \( 1 + (0.461 - 0.887i)T \)
89 \( 1 + (0.258 - 0.965i)T \)
97 \( 1 + (0.939 + 0.342i)T \)
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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

−19.95338942059668239841201202312, −19.15465616988019149509704082755, −18.713470119301479065237745522863, −17.91646522490866144161056631938, −16.600587007071794099542821650905, −16.41055246486243560269851239539, −15.73051619132111307264902368725, −14.734219677011584790681622860898, −14.11533069507242883154559241342, −12.97791833965605109039604940576, −12.783691299291331152484383504460, −11.58701498389523368720240599639, −10.853736446198795345739402279033, −10.41772348907107066835754480435, −9.32939042584874334804433024057, −8.28331751061770874461978354672, −7.8985587117931549221501735630, −6.92617421744559394214685528738, −6.188231383113737243427586249687, −5.16186966591202922605928196201, −4.17200152888666532256064226428, −3.42744862799763635670416657932, −2.85237271211631524977540592018, −1.16904798920518468652456216189, −0.45291854497218469376535645495, 0.605865403961076426240836653291, 1.92872056138615038934573607890, 2.88627971189508874102080971732, 3.75777016837433233743637674683, 4.50967206649107444037368411652, 5.59143919773437975179993181439, 6.26550292771050659366936116786, 7.37099903266297738697097834791, 7.901885693420451447786445024922, 8.87580684607283639970599203195, 9.453055290636563255352240831491, 10.52776156257432366742024778469, 11.27877052782834214492158993602, 11.954971509423515667955485403419, 12.89846050627466599547165213104, 13.11684948745250137649267310523, 14.52966047971517403004179518310, 15.23341664160027767276330765093, 15.64608733997760444918266129696, 16.38498261301688147300930802057, 17.227424735309301818465238103105, 18.20949870727127388181038014596, 18.921092694616692103533793630907, 19.27847894043916731402340161414, 20.191304413577818583043600385207

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