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Properties

Label 1-539-539.113-r0-0-0
Degree $1$
Conductor $539$
Sign $-0.186 + 0.982i$
Analytic cond. $2.50310$
Root an. cond. $2.50310$
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  + (−0.691 + 0.722i)2-s + (0.753 + 0.657i)3-s + (−0.0448 − 0.998i)4-s + (0.858 − 0.512i)5-s + (−0.995 + 0.0896i)6-s + (0.753 + 0.657i)8-s + (0.134 + 0.990i)9-s + (−0.222 + 0.974i)10-s + (0.623 − 0.781i)12-s + (−0.691 + 0.722i)13-s + (0.983 + 0.178i)15-s + (−0.995 + 0.0896i)16-s + (0.936 + 0.351i)17-s + (−0.809 − 0.587i)18-s + (−0.809 + 0.587i)19-s + (−0.550 − 0.834i)20-s + ⋯
L(s)  = 1  + (−0.691 + 0.722i)2-s + (0.753 + 0.657i)3-s + (−0.0448 − 0.998i)4-s + (0.858 − 0.512i)5-s + (−0.995 + 0.0896i)6-s + (0.753 + 0.657i)8-s + (0.134 + 0.990i)9-s + (−0.222 + 0.974i)10-s + (0.623 − 0.781i)12-s + (−0.691 + 0.722i)13-s + (0.983 + 0.178i)15-s + (−0.995 + 0.0896i)16-s + (0.936 + 0.351i)17-s + (−0.809 − 0.587i)18-s + (−0.809 + 0.587i)19-s + (−0.550 − 0.834i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.186 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.186 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $-0.186 + 0.982i$
Analytic conductor: \(2.50310\)
Root analytic conductor: \(2.50310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (0:\ ),\ -0.186 + 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8923463948 + 1.077901006i\)
\(L(\frac12)\) \(\approx\) \(0.8923463948 + 1.077901006i\)
\(L(1)\) \(\approx\) \(0.9438549110 + 0.5631966804i\)
\(L(1)\) \(\approx\) \(0.9438549110 + 0.5631966804i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.691 + 0.722i)T \)
3 \( 1 + (0.753 + 0.657i)T \)
5 \( 1 + (0.858 - 0.512i)T \)
13 \( 1 + (-0.691 + 0.722i)T \)
17 \( 1 + (0.936 + 0.351i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + (0.623 + 0.781i)T \)
29 \( 1 + (-0.963 + 0.266i)T \)
31 \( 1 + (0.309 - 0.951i)T \)
37 \( 1 + (-0.963 + 0.266i)T \)
41 \( 1 + (0.753 + 0.657i)T \)
43 \( 1 + (-0.222 + 0.974i)T \)
47 \( 1 + (0.983 - 0.178i)T \)
53 \( 1 + (0.936 - 0.351i)T \)
59 \( 1 + (0.753 - 0.657i)T \)
61 \( 1 + (0.936 + 0.351i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.550 + 0.834i)T \)
73 \( 1 + (0.983 + 0.178i)T \)
79 \( 1 + (0.309 - 0.951i)T \)
83 \( 1 + (-0.691 - 0.722i)T \)
89 \( 1 + (-0.900 + 0.433i)T \)
97 \( 1 + (0.309 - 0.951i)T \)
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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

−22.98763858023448757557701029335, −22.19350461758377520010816184962, −21.14738999491075228894683578423, −20.71210262458374361633627582378, −19.677166151362350483526161271564, −18.9815284876420150297615468470, −18.34209002635549035122682957950, −17.49533642910600180486278038912, −16.92379186999367839610457077677, −15.4268216654204195863991576, −14.47369832664809079594264646239, −13.65566919033262121908282103516, −12.77653402486905310028186992957, −12.18191836021423633487474936191, −10.84786400711888130164370058782, −10.11528559502748828742594373950, −9.243852601925350783213630279389, −8.4821043488695707464306486434, −7.37072641945071094680524183657, −6.80448222152030650171860984508, −5.36884099732443703408762487849, −3.74989468539960299217879785535, −2.71844860047005535010816461842, −2.1912620604624667350739486351, −0.89566744611192273476495687531, 1.48162462271723499886935501325, 2.3824337430380395031935460045, 4.00873048460834886686481851325, 5.07201411645229427065939410387, 5.82191769793648869355065310146, 7.073480543236864144842895068673, 8.07244745515643853422765484204, 8.86836455579289877194030872188, 9.68893731822663953527380295773, 10.098090635599493823972902634667, 11.2648559416127679810852018582, 12.79206197297866930560123283643, 13.73666739543956905617832078186, 14.53653831731806904439776083179, 15.07219412222256852106661964933, 16.278436997083700984023222249565, 16.804821907762365175868901438862, 17.458709632490597783392014499625, 18.79537096520925178516928610274, 19.26475695668340407088811995123, 20.31853246920527227790991796232, 21.07119540192723582027999438721, 21.78243258869677999472591193365, 22.91512014352292365992312516299, 24.01921551270751476009087827122

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