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Properties

Label 1-4729-4729.454-r0-0-0
Degree $1$
Conductor $4729$
Sign $-0.533 - 0.845i$
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  + (−0.150 − 0.988i)2-s + (0.901 − 0.431i)3-s + (−0.954 + 0.298i)4-s + (−0.395 + 0.918i)5-s + (−0.563 − 0.826i)6-s + (0.975 + 0.221i)7-s + (0.439 + 0.898i)8-s + (0.627 − 0.778i)9-s + (0.967 + 0.252i)10-s + (0.336 + 0.941i)11-s + (−0.732 + 0.681i)12-s + (−0.182 − 0.983i)13-s + (0.0717 − 0.997i)14-s + (0.0398 + 0.999i)15-s + (0.821 − 0.569i)16-s + (−0.380 − 0.924i)17-s + ⋯
L(s)  = 1  + (−0.150 − 0.988i)2-s + (0.901 − 0.431i)3-s + (−0.954 + 0.298i)4-s + (−0.395 + 0.918i)5-s + (−0.563 − 0.826i)6-s + (0.975 + 0.221i)7-s + (0.439 + 0.898i)8-s + (0.627 − 0.778i)9-s + (0.967 + 0.252i)10-s + (0.336 + 0.941i)11-s + (−0.732 + 0.681i)12-s + (−0.182 − 0.983i)13-s + (0.0717 − 0.997i)14-s + (0.0398 + 0.999i)15-s + (0.821 − 0.569i)16-s + (−0.380 − 0.924i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.533 - 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.533 - 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $-0.533 - 0.845i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (454, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ -0.533 - 0.845i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9769850347 - 1.771053897i\)
\(L(\frac12)\) \(\approx\) \(0.9769850347 - 1.771053897i\)
\(L(1)\) \(\approx\) \(1.086731668 - 0.6566455275i\)
\(L(1)\) \(\approx\) \(1.086731668 - 0.6566455275i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (-0.150 - 0.988i)T \)
3 \( 1 + (0.901 - 0.431i)T \)
5 \( 1 + (-0.395 + 0.918i)T \)
7 \( 1 + (0.975 + 0.221i)T \)
11 \( 1 + (0.336 + 0.941i)T \)
13 \( 1 + (-0.182 - 0.983i)T \)
17 \( 1 + (-0.380 - 0.924i)T \)
19 \( 1 + (-0.275 + 0.961i)T \)
23 \( 1 + (0.753 - 0.657i)T \)
29 \( 1 + (0.150 - 0.988i)T \)
31 \( 1 + (0.213 + 0.976i)T \)
37 \( 1 + (0.959 - 0.283i)T \)
41 \( 1 + (-0.927 - 0.373i)T \)
43 \( 1 + (0.0875 - 0.996i)T \)
47 \( 1 + (-0.351 - 0.936i)T \)
53 \( 1 + (-0.522 - 0.852i)T \)
59 \( 1 + (-0.639 - 0.768i)T \)
61 \( 1 + (-0.803 + 0.595i)T \)
67 \( 1 + (-0.894 - 0.446i)T \)
71 \( 1 + (0.856 - 0.516i)T \)
73 \( 1 + (0.675 - 0.737i)T \)
79 \( 1 - T \)
83 \( 1 + (-0.999 - 0.0159i)T \)
89 \( 1 + (-0.198 - 0.980i)T \)
97 \( 1 + (0.975 + 0.221i)T \)
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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

−18.4198995537989352241543187485, −17.40645120318143064829483099896, −16.83444142069109335066527643371, −16.497323797801742283349030869833, −15.549474059926144705112105331195, −15.179135075347680019965012178731, −14.4567119776939508295955948374, −13.865982862897410290589660515678, −13.27269411594245767569085350493, −12.677955873573715210881536477231, −11.38427903862545801605270789782, −11.02801426557286434451304190126, −9.89362067808185560232406338412, −9.03604874876319198983636843423, −8.881301164105021382963203133742, −8.12587803541334202885011640876, −7.65273842896265816742920011312, −6.80699915558391656568273407588, −5.88783939717929104173538398855, −4.91288932876370902607842389642, −4.48759639861740587886906191839, −3.993595495291305985370821378583, −2.99397078884792975156046400350, −1.62701014235066130615492126763, −1.11676833986672497706878045177, 0.53799356340087727206485775015, 1.6891321159182552119758002877, 2.21948385783858045744981467934, 2.91331518011695767207259665791, 3.58112225256365822862480483689, 4.422293016386012231144951507451, 5.03588307785909419541624048323, 6.28720806341533364736239023142, 7.26312994294777533925455595636, 7.71191757886352649245575835975, 8.38804275148959302482490990788, 9.00840140551220421543047752229, 9.978717210602653014765168937569, 10.34316427056487658589297012561, 11.221564838368347931602533657124, 11.94428576816928915262934654244, 12.36738912408537510441195605468, 13.16062773553306872232336748605, 14.02459550661548173723904665546, 14.40420184300287063609851505279, 15.09284405096489911267463063337, 15.50001008707251584215592878162, 16.929208879942439341210725818764, 17.666543613095491180877514123073, 18.23886573483272074426356273293

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