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Properties

Label 1-4729-4729.1082-r0-0-0
Degree $1$
Conductor $4729$
Sign $-0.536 - 0.844i$
Analytic cond. $21.9613$
Root an. cond. $21.9613$
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  + (−0.949 − 0.313i)2-s + (−0.927 + 0.373i)3-s + (0.803 + 0.595i)4-s + (−0.270 + 0.962i)5-s + (0.997 − 0.0637i)6-s + (0.326 + 0.945i)7-s + (−0.576 − 0.817i)8-s + (0.721 − 0.692i)9-s + (0.558 − 0.829i)10-s + (0.868 − 0.495i)11-s + (−0.967 − 0.252i)12-s + (−0.904 − 0.427i)13-s + (−0.0132 − 0.999i)14-s + (−0.108 − 0.994i)15-s + (0.290 + 0.956i)16-s + (−0.653 + 0.756i)17-s + ⋯
L(s)  = 1  + (−0.949 − 0.313i)2-s + (−0.927 + 0.373i)3-s + (0.803 + 0.595i)4-s + (−0.270 + 0.962i)5-s + (0.997 − 0.0637i)6-s + (0.326 + 0.945i)7-s + (−0.576 − 0.817i)8-s + (0.721 − 0.692i)9-s + (0.558 − 0.829i)10-s + (0.868 − 0.495i)11-s + (−0.967 − 0.252i)12-s + (−0.904 − 0.427i)13-s + (−0.0132 − 0.999i)14-s + (−0.108 − 0.994i)15-s + (0.290 + 0.956i)16-s + (−0.653 + 0.756i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.536 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.536 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $-0.536 - 0.844i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (1082, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ -0.536 - 0.844i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1403433142 + 0.2554380570i\)
\(L(\frac12)\) \(\approx\) \(-0.1403433142 + 0.2554380570i\)
\(L(1)\) \(\approx\) \(0.4093045838 + 0.2024057278i\)
\(L(1)\) \(\approx\) \(0.4093045838 + 0.2024057278i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (-0.949 - 0.313i)T \)
3 \( 1 + (-0.927 + 0.373i)T \)
5 \( 1 + (-0.270 + 0.962i)T \)
7 \( 1 + (0.326 + 0.945i)T \)
11 \( 1 + (0.868 - 0.495i)T \)
13 \( 1 + (-0.904 - 0.427i)T \)
17 \( 1 + (-0.653 + 0.756i)T \)
19 \( 1 + (0.280 + 0.959i)T \)
23 \( 1 + (-0.924 + 0.380i)T \)
29 \( 1 + (0.313 + 0.949i)T \)
31 \( 1 + (-0.999 - 0.0398i)T \)
37 \( 1 + (-0.545 + 0.838i)T \)
41 \( 1 + (0.994 + 0.103i)T \)
43 \( 1 + (0.0743 + 0.997i)T \)
47 \( 1 + (-0.997 - 0.0663i)T \)
53 \( 1 + (-0.272 + 0.962i)T \)
59 \( 1 + (-0.735 + 0.677i)T \)
61 \( 1 + (-0.641 - 0.767i)T \)
67 \( 1 + (0.979 - 0.200i)T \)
71 \( 1 + (-0.486 + 0.873i)T \)
73 \( 1 + (-0.531 + 0.846i)T \)
79 \( 1 + (0.866 + 0.5i)T \)
83 \( 1 + (0.833 - 0.551i)T \)
89 \( 1 + (0.0212 + 0.999i)T \)
97 \( 1 + (-0.326 - 0.945i)T \)
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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

−17.73583398952471750736805976594, −17.11875630968221503629995610064, −16.524664508321232429776557195915, −16.09711199331499177586304323494, −15.353625553796504965952795789161, −14.37163506510457734093800280728, −13.74276342284125955821878552135, −12.83462996242196847596113317246, −12.01125897139309804208036165026, −11.64555461226183456824116023426, −10.98426054299007933850190843820, −10.17709320979822820082348933988, −9.4430038853349255019434056379, −8.9747229100567789304059650428, −7.84035560326458105887277339935, −7.4325892238638244131395582490, −6.82827682772062111841515167447, −6.15842839175188667928450925733, −5.04886155818884256611800086448, −4.72044452946347117046556097892, −3.878892591118947373017283051184, −2.19317597677716883556791478385, −1.71880561270592047689644186614, −0.69041349388206242382507074271, −0.18204988430393649384703694490, 1.32236855567404512502465938602, 2.00750897691968281587944618271, 3.06042261321168018377445982517, 3.64579688457892909777644188161, 4.55794781744411420253647527827, 5.75649268907535619928790550460, 6.15134815269051404272588870089, 6.87634549315609215496586042726, 7.680811614925141534633190131841, 8.35905696014191282753910897819, 9.2669932990122439193082322446, 9.831170047923058652900752214066, 10.529260194365945771255414797986, 11.15258474519049206774599583154, 11.61383967000023473186419155785, 12.286521684743352251382296183560, 12.69702377479759415930281314660, 14.20114716000565860099312955349, 14.823257167012663093282327388880, 15.4076282325611760077530858740, 16.0386961518144755447353900189, 16.723544656410913637689846051088, 17.40756221933844041238991962583, 18.054562189011696977380856158192, 18.33515254557094451581942604136

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