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Properties

Label 1-4729-4729.486-r0-0-0
Degree $1$
Conductor $4729$
Sign $-0.999 + 0.00863i$
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  + (−0.933 − 0.358i)2-s + (−0.481 + 0.876i)3-s + (0.742 + 0.669i)4-s + (−0.793 + 0.608i)5-s + (0.763 − 0.645i)6-s + (−0.509 + 0.860i)7-s + (−0.453 − 0.891i)8-s + (−0.536 − 0.843i)9-s + (0.959 − 0.283i)10-s + (−0.549 + 0.835i)11-s + (−0.944 + 0.328i)12-s + (0.927 − 0.373i)13-s + (0.783 − 0.620i)14-s + (−0.150 − 0.988i)15-s + (0.103 + 0.994i)16-s + (−0.856 + 0.516i)17-s + ⋯
L(s)  = 1  + (−0.933 − 0.358i)2-s + (−0.481 + 0.876i)3-s + (0.742 + 0.669i)4-s + (−0.793 + 0.608i)5-s + (0.763 − 0.645i)6-s + (−0.509 + 0.860i)7-s + (−0.453 − 0.891i)8-s + (−0.536 − 0.843i)9-s + (0.959 − 0.283i)10-s + (−0.549 + 0.835i)11-s + (−0.944 + 0.328i)12-s + (0.927 − 0.373i)13-s + (0.783 − 0.620i)14-s + (−0.150 − 0.988i)15-s + (0.103 + 0.994i)16-s + (−0.856 + 0.516i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00863i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00863i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $-0.999 + 0.00863i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (486, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ -0.999 + 0.00863i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.001574051935 + 0.3647169066i\)
\(L(\frac12)\) \(\approx\) \(0.001574051935 + 0.3647169066i\)
\(L(1)\) \(\approx\) \(0.4228972318 + 0.1857310629i\)
\(L(1)\) \(\approx\) \(0.4228972318 + 0.1857310629i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (-0.933 - 0.358i)T \)
3 \( 1 + (-0.481 + 0.876i)T \)
5 \( 1 + (-0.793 + 0.608i)T \)
7 \( 1 + (-0.509 + 0.860i)T \)
11 \( 1 + (-0.549 + 0.835i)T \)
13 \( 1 + (0.927 - 0.373i)T \)
17 \( 1 + (-0.856 + 0.516i)T \)
19 \( 1 + (0.872 - 0.488i)T \)
23 \( 1 + (-0.915 - 0.402i)T \)
29 \( 1 + (0.933 - 0.358i)T \)
31 \( 1 + (0.424 - 0.905i)T \)
37 \( 1 + (-0.894 + 0.446i)T \)
41 \( 1 + (0.908 - 0.417i)T \)
43 \( 1 + (-0.290 - 0.956i)T \)
47 \( 1 + (0.978 + 0.205i)T \)
53 \( 1 + (-0.410 + 0.912i)T \)
59 \( 1 + (0.981 + 0.190i)T \)
61 \( 1 + (0.753 + 0.657i)T \)
67 \( 1 + (0.991 - 0.127i)T \)
71 \( 1 + (-0.984 + 0.174i)T \)
73 \( 1 + (0.803 - 0.595i)T \)
79 \( 1 - T \)
83 \( 1 + (-0.366 + 0.930i)T \)
89 \( 1 + (0.687 + 0.726i)T \)
97 \( 1 + (-0.509 + 0.860i)T \)
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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

−17.825847689564153757649168240956, −17.260494688296160198153554720018, −16.27928291643829576171007352551, −16.02580171512154574629188138233, −15.84651506342313348146183168466, −14.25567089755124163505644545679, −13.85283999622974047910561664988, −13.11377903434553917074249450720, −12.324410386004225397195306742830, −11.49483840600003187205488146976, −11.17105733251913223663762326487, −10.42167221006471921605770728255, −9.5967706699513664490249679100, −8.50662312663864797778715204009, −8.3523368741164323395276239379, −7.45668050259584226295661300968, −6.96541986421377590525126922230, −6.24765506407829405965590628641, −5.53947708632580870930462842093, −4.72414211809271817999246416858, −3.61987506603624635377636303617, −2.802598443901412922043851724883, −1.5872282635350489663703147657, −0.95221836046601064483507319568, −0.232415056671864920542876920386, 0.8196656282781665200837647932, 2.35086572913637434729309126134, 2.747310346719852386919510026437, 3.73267767785232851099361802045, 4.19048989249832716605198983029, 5.34084132345279501825130935535, 6.23105259082075340596875741119, 6.71256887396384087987427257432, 7.664179135886839709426508610697, 8.441058029926481384484873170176, 8.95321640044889209437514424933, 9.82966306667556699322787579469, 10.34855339138749208384075276847, 10.88937000475894932829570715633, 11.65697801160929578484342018254, 12.0857810332615807742015280199, 12.746931853108036434565421263125, 13.786037311605070792304362352686, 15.04794714743106043102344662027, 15.42214770259045079683113461070, 15.86181086882390673426409024996, 16.219308647819324426117236071012, 17.49470299370348003488076868227, 17.72005006592069301919605342795, 18.50437729046870601654745664302

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