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Properties

Label 1-4729-4729.1042-r0-0-0
Degree $1$
Conductor $4729$
Sign $-0.159 - 0.987i$
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  + (0.830 − 0.556i)2-s + (0.996 + 0.0796i)3-s + (0.380 − 0.924i)4-s + (0.580 − 0.814i)5-s + (0.872 − 0.488i)6-s + (0.885 + 0.465i)7-s + (−0.198 − 0.980i)8-s + (0.987 + 0.158i)9-s + (0.0292 − 0.999i)10-s + (0.843 + 0.536i)11-s + (0.453 − 0.891i)12-s + (−0.925 − 0.378i)13-s + (0.994 − 0.106i)14-s + (0.643 − 0.765i)15-s + (−0.709 − 0.704i)16-s + (0.551 + 0.833i)17-s + ⋯
L(s)  = 1  + (0.830 − 0.556i)2-s + (0.996 + 0.0796i)3-s + (0.380 − 0.924i)4-s + (0.580 − 0.814i)5-s + (0.872 − 0.488i)6-s + (0.885 + 0.465i)7-s + (−0.198 − 0.980i)8-s + (0.987 + 0.158i)9-s + (0.0292 − 0.999i)10-s + (0.843 + 0.536i)11-s + (0.453 − 0.891i)12-s + (−0.925 − 0.378i)13-s + (0.994 − 0.106i)14-s + (0.643 − 0.765i)15-s + (−0.709 − 0.704i)16-s + (0.551 + 0.833i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.159 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.159 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $-0.159 - 0.987i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (1042, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ -0.159 - 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.847605352 - 4.520955282i\)
\(L(\frac12)\) \(\approx\) \(3.847605352 - 4.520955282i\)
\(L(1)\) \(\approx\) \(2.517297791 - 1.450382720i\)
\(L(1)\) \(\approx\) \(2.517297791 - 1.450382720i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (0.830 - 0.556i)T \)
3 \( 1 + (0.996 + 0.0796i)T \)
5 \( 1 + (0.580 - 0.814i)T \)
7 \( 1 + (0.885 + 0.465i)T \)
11 \( 1 + (0.843 + 0.536i)T \)
13 \( 1 + (-0.925 - 0.378i)T \)
17 \( 1 + (0.551 + 0.833i)T \)
19 \( 1 + (-0.647 - 0.762i)T \)
23 \( 1 + (0.0159 - 0.999i)T \)
29 \( 1 + (0.556 - 0.830i)T \)
31 \( 1 + (0.313 - 0.949i)T \)
37 \( 1 + (-0.0981 - 0.995i)T \)
41 \( 1 + (-0.737 + 0.675i)T \)
43 \( 1 + (0.560 + 0.827i)T \)
47 \( 1 + (-0.506 + 0.862i)T \)
53 \( 1 + (-0.801 - 0.597i)T \)
59 \( 1 + (-0.946 - 0.323i)T \)
61 \( 1 + (-0.653 + 0.756i)T \)
67 \( 1 + (-0.0451 - 0.998i)T \)
71 \( 1 + (0.606 + 0.795i)T \)
73 \( 1 + (0.224 + 0.974i)T \)
79 \( 1 + (-0.866 + 0.5i)T \)
83 \( 1 + (-0.0345 + 0.999i)T \)
89 \( 1 + (0.169 + 0.985i)T \)
97 \( 1 + (-0.885 - 0.465i)T \)
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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

−18.34482031368744329338868653285, −17.4703206346462483842739596897, −17.0373769905532021438670043899, −16.25291737800450082662783793804, −15.31562612353905350832788710294, −14.77697931602822370301480283783, −14.170574364888727408813502839186, −13.978733557397418947167579578486, −13.4334310368725264315892783189, −12.24404163816292746915404777265, −11.897936024548513142214118839584, −10.90508564437743945571728652187, −10.21223822084087540234329469403, −9.35573660589236645048186613963, −8.607990170292082097639419840810, −7.88991120916219162897839745029, −7.116575990159609732860932887463, −6.8741677346020618748773105123, −5.89034393313376164248886299777, −4.994846458565216965889740984842, −4.36492601374146879199590866743, −3.347258480808137730905392158910, −3.12312480411000714891173243131, −1.960830715452702758055852260582, −1.51902595190669515296895184749, 0.96188960524453459555757036491, 1.740342551590427712225365199746, 2.29651871962383121526776450120, 2.90560737559563191392223268697, 4.27802138882290340337738794501, 4.374321068854264724819014365999, 5.148600573157796685285135650534, 6.056301992780778128374017466409, 6.7651456444974098907682964641, 7.878528342803695303528898966967, 8.39164426927175040948516588574, 9.37552304339738909969509281687, 9.65220754646221770091283198908, 10.47109130238386593209989320529, 11.30513943032601103594137597605, 12.32781541647942598376617597519, 12.51687292572905244054681375775, 13.202204376496264211634010576433, 14.11488696156646294616093683409, 14.463882749017252768953299261948, 15.07128735953986489958445130813, 15.55296813151898980467821057537, 16.63762079788821355894089590946, 17.32437082059637738369775560652, 18.02212683563977982579252224942

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