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Properties

Label 1-4729-4729.189-r0-0-0
Degree $1$
Conductor $4729$
Sign $-0.919 - 0.392i$
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  + (−0.509 + 0.860i)2-s + (0.321 + 0.947i)3-s + (−0.481 − 0.876i)4-s + (0.476 + 0.879i)5-s + (−0.978 − 0.205i)6-s + (−0.0981 − 0.995i)7-s + (0.999 + 0.0318i)8-s + (−0.793 + 0.608i)9-s + (−0.999 − 0.0372i)10-s + (0.275 + 0.961i)11-s + (0.675 − 0.737i)12-s + (0.796 − 0.604i)13-s + (0.906 + 0.422i)14-s + (−0.679 + 0.733i)15-s + (−0.536 + 0.843i)16-s + (−0.735 + 0.677i)17-s + ⋯
L(s)  = 1  + (−0.509 + 0.860i)2-s + (0.321 + 0.947i)3-s + (−0.481 − 0.876i)4-s + (0.476 + 0.879i)5-s + (−0.978 − 0.205i)6-s + (−0.0981 − 0.995i)7-s + (0.999 + 0.0318i)8-s + (−0.793 + 0.608i)9-s + (−0.999 − 0.0372i)10-s + (0.275 + 0.961i)11-s + (0.675 − 0.737i)12-s + (0.796 − 0.604i)13-s + (0.906 + 0.422i)14-s + (−0.679 + 0.733i)15-s + (−0.536 + 0.843i)16-s + (−0.735 + 0.677i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.919 - 0.392i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.919 - 0.392i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $-0.919 - 0.392i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ -0.919 - 0.392i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3016660727 + 1.477084457i\)
\(L(\frac12)\) \(\approx\) \(-0.3016660727 + 1.477084457i\)
\(L(1)\) \(\approx\) \(0.5989674692 + 0.7833822855i\)
\(L(1)\) \(\approx\) \(0.5989674692 + 0.7833822855i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (-0.509 + 0.860i)T \)
3 \( 1 + (0.321 + 0.947i)T \)
5 \( 1 + (0.476 + 0.879i)T \)
7 \( 1 + (-0.0981 - 0.995i)T \)
11 \( 1 + (0.275 + 0.961i)T \)
13 \( 1 + (0.796 - 0.604i)T \)
17 \( 1 + (-0.735 + 0.677i)T \)
19 \( 1 + (0.862 + 0.506i)T \)
23 \( 1 + (0.639 + 0.768i)T \)
29 \( 1 + (0.509 + 0.860i)T \)
31 \( 1 + (-0.260 + 0.965i)T \)
37 \( 1 + (-0.767 + 0.641i)T \)
41 \( 1 + (0.0557 - 0.998i)T \)
43 \( 1 + (-0.239 + 0.970i)T \)
47 \( 1 + (0.571 + 0.820i)T \)
53 \( 1 + (0.622 - 0.782i)T \)
59 \( 1 + (0.749 - 0.661i)T \)
61 \( 1 + (-0.0928 - 0.995i)T \)
67 \( 1 + (0.610 - 0.792i)T \)
71 \( 1 + (-0.951 - 0.308i)T \)
73 \( 1 + (-0.140 + 0.990i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.946 + 0.323i)T \)
89 \( 1 + (0.997 + 0.0690i)T \)
97 \( 1 + (-0.0981 - 0.995i)T \)
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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

−17.93928280564199945279988748029, −17.42655700885005356530226469544, −16.497158613461432788152382338562, −16.129132154459595837119747730705, −15.06746108031245499500931176327, −13.939719642635022369128153650806, −13.506232727718452646811701471255, −13.16148209181220031545932767947, −12.25341162988734404370430648708, −11.68262635849990697490185405733, −11.38588484082437034892579406638, −10.2551809115796985329195073537, −9.10839978984354967894777108371, −9.006889793691198905695051803794, −8.5825660733580015191264371246, −7.69936739483829589304728973124, −6.769889373195935582047369966517, −6.00832333880530263258054178811, −5.280892112723487083193946427246, −4.30734386043302344632474332255, −3.37111152560752011389027964532, −2.51474365889401157151668306077, −2.098754744709560589246868409273, −1.1006592915546249935026891501, −0.52874131559232314828092107346, 1.19587642108644378544967943502, 1.98354814649747003522767340799, 3.32030066921230619523904868624, 3.703891177751375832916404375426, 4.73164242982012200000776384934, 5.322831060389467341745725238809, 6.212177975900377554734280745915, 6.8998366092864002195947501434, 7.4700135423948210191814503539, 8.28693223619852523388371813275, 9.0374735558655532659921168644, 9.7824368518966536684871038501, 10.216367731196014555463071419942, 10.77979470637655181225695429758, 11.281673845511936483304874192, 12.75860869109048988007046852117, 13.566967684167518089005371625438, 14.10816347105578707461147530142, 14.56352655743520667483343788330, 15.37105823797591173570879169137, 15.71280144852484068564013001007, 16.50035315293524653324173985501, 17.31798187920810151153479107331, 17.631898869409054354123605566989, 18.24804621042667642408176618883

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