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Properties

Label 1-511-511.479-r1-0-0
Degree $1$
Conductor $511$
Sign $-0.712 + 0.702i$
Analytic cond. $54.9145$
Root an. cond. $54.9145$
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  + (0.173 − 0.984i)2-s − 3-s + (−0.939 − 0.342i)4-s + (0.766 − 0.642i)5-s + (−0.173 + 0.984i)6-s + (−0.5 + 0.866i)8-s + 9-s + (−0.5 − 0.866i)10-s + (−0.173 − 0.984i)11-s + (0.939 + 0.342i)12-s + (0.173 − 0.984i)13-s + (−0.766 + 0.642i)15-s + (0.766 + 0.642i)16-s + (−0.5 + 0.866i)17-s + (0.173 − 0.984i)18-s + (−0.173 + 0.984i)19-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)2-s − 3-s + (−0.939 − 0.342i)4-s + (0.766 − 0.642i)5-s + (−0.173 + 0.984i)6-s + (−0.5 + 0.866i)8-s + 9-s + (−0.5 − 0.866i)10-s + (−0.173 − 0.984i)11-s + (0.939 + 0.342i)12-s + (0.173 − 0.984i)13-s + (−0.766 + 0.642i)15-s + (0.766 + 0.642i)16-s + (−0.5 + 0.866i)17-s + (0.173 − 0.984i)18-s + (−0.173 + 0.984i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 511 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.712 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 511 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.712 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(511\)    =    \(7 \cdot 73\)
Sign: $-0.712 + 0.702i$
Analytic conductor: \(54.9145\)
Root analytic conductor: \(54.9145\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{511} (479, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 511,\ (1:\ ),\ -0.712 + 0.702i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4037072406 - 0.9844049780i\)
\(L(\frac12)\) \(\approx\) \(-0.4037072406 - 0.9844049780i\)
\(L(1)\) \(\approx\) \(0.5645893856 - 0.6277469248i\)
\(L(1)\) \(\approx\) \(0.5645893856 - 0.6277469248i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
73 \( 1 \)
good2 \( 1 + (0.173 - 0.984i)T \)
3 \( 1 - T \)
5 \( 1 + (0.766 - 0.642i)T \)
11 \( 1 + (-0.173 - 0.984i)T \)
13 \( 1 + (0.173 - 0.984i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-0.173 + 0.984i)T \)
23 \( 1 + (0.766 - 0.642i)T \)
29 \( 1 + (0.939 - 0.342i)T \)
31 \( 1 + (0.173 - 0.984i)T \)
37 \( 1 + (0.766 - 0.642i)T \)
41 \( 1 + (-0.173 - 0.984i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (-0.939 - 0.342i)T \)
53 \( 1 + (-0.766 - 0.642i)T \)
59 \( 1 + (-0.939 + 0.342i)T \)
61 \( 1 + (0.939 + 0.342i)T \)
67 \( 1 + (-0.939 + 0.342i)T \)
71 \( 1 + (-0.939 + 0.342i)T \)
79 \( 1 + (0.766 + 0.642i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.766 - 0.642i)T \)
97 \( 1 + (0.5 + 0.866i)T \)
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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

−23.64312503859814219492027335954, −23.236348301120571170385107580699, −22.33896164906187158452119382267, −21.71497085132566886278663485442, −21.023761738440978284590885123701, −19.37129510374163737594962777703, −18.287011216211266787142321231, −17.83874887461848326135296831351, −17.21078285592367865599350904463, −16.24545445921621906434191290922, −15.48324009170516803157099654802, −14.59571251518699490537134807919, −13.596363790449646446016061433876, −12.94198509074090350221695390105, −11.81036022624023036715175981668, −10.85475446831727432067612054033, −9.72084008256845847599180034755, −9.17687110099968011296758424931, −7.533373597083365741134088238652, −6.71372750830101918794378074514, −6.35181945927423695682931685756, −4.954305494932122033283352317798, −4.62695363642288369869914100632, −2.93147865793428672407310761333, −1.357871227882573725466604550906, 0.34025184998974985438696260859, 1.15878067427411252584268794540, 2.34366286790021931652356144384, 3.75329854703912314197651160849, 4.77678021325046721362261642425, 5.718774441886678295088206871586, 6.185990047342132923262857205625, 8.08094449988588278444213179907, 8.93497973434430845778117877905, 10.15200780499037585097630982231, 10.56967729294068095800523729635, 11.51447335732420145905529097410, 12.591759488632572069596380379966, 12.986741172221645691892707835628, 13.86618947336579055175586531439, 15.08627627481848407779211465441, 16.291294385243292354092079553433, 17.140256748742049924768404494205, 17.76676578398209884935023445723, 18.61881293898463178835173439490, 19.44233696053043309389800533907, 20.6955291631635286403017261896, 21.124577166126756079253386206487, 21.99514511903016001024528369313, 22.605945914896704699028761269067

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