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Properties

Label 1-4729-4729.655-r0-0-0
Degree $1$
Conductor $4729$
Sign $0.443 + 0.896i$
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  + (−0.763 + 0.645i)2-s + (−0.915 + 0.402i)3-s + (0.166 − 0.986i)4-s + (0.760 − 0.649i)5-s + (0.439 − 0.898i)6-s + (−0.311 − 0.950i)7-s + (0.509 + 0.860i)8-s + (0.675 − 0.737i)9-s + (−0.161 + 0.986i)10-s + (0.417 + 0.908i)11-s + (0.244 + 0.969i)12-s + (−0.991 + 0.132i)13-s + (0.851 + 0.525i)14-s + (−0.434 + 0.900i)15-s + (−0.944 − 0.328i)16-s + (−0.100 − 0.994i)17-s + ⋯
L(s)  = 1  + (−0.763 + 0.645i)2-s + (−0.915 + 0.402i)3-s + (0.166 − 0.986i)4-s + (0.760 − 0.649i)5-s + (0.439 − 0.898i)6-s + (−0.311 − 0.950i)7-s + (0.509 + 0.860i)8-s + (0.675 − 0.737i)9-s + (−0.161 + 0.986i)10-s + (0.417 + 0.908i)11-s + (0.244 + 0.969i)12-s + (−0.991 + 0.132i)13-s + (0.851 + 0.525i)14-s + (−0.434 + 0.900i)15-s + (−0.944 − 0.328i)16-s + (−0.100 − 0.994i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.443 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.443 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $0.443 + 0.896i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (655, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ 0.443 + 0.896i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6995872009 + 0.4346318086i\)
\(L(\frac12)\) \(\approx\) \(0.6995872009 + 0.4346318086i\)
\(L(1)\) \(\approx\) \(0.6161912205 + 0.1361406770i\)
\(L(1)\) \(\approx\) \(0.6161912205 + 0.1361406770i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (-0.763 + 0.645i)T \)
3 \( 1 + (-0.915 + 0.402i)T \)
5 \( 1 + (0.760 - 0.649i)T \)
7 \( 1 + (-0.311 - 0.950i)T \)
11 \( 1 + (0.417 + 0.908i)T \)
13 \( 1 + (-0.991 + 0.132i)T \)
17 \( 1 + (-0.100 - 0.994i)T \)
19 \( 1 + (0.867 + 0.497i)T \)
23 \( 1 + (0.973 + 0.229i)T \)
29 \( 1 + (-0.645 + 0.763i)T \)
31 \( 1 + (-0.996 + 0.0875i)T \)
37 \( 1 + (0.993 - 0.116i)T \)
41 \( 1 + (0.516 + 0.856i)T \)
43 \( 1 + (-0.964 - 0.265i)T \)
47 \( 1 + (0.368 - 0.929i)T \)
53 \( 1 + (0.992 + 0.124i)T \)
59 \( 1 + (-0.897 + 0.441i)T \)
61 \( 1 + (0.928 + 0.370i)T \)
67 \( 1 + (0.923 - 0.383i)T \)
71 \( 1 + (-0.970 - 0.242i)T \)
73 \( 1 + (0.385 + 0.922i)T \)
79 \( 1 + (0.866 + 0.5i)T \)
83 \( 1 + (0.690 - 0.722i)T \)
89 \( 1 + (0.363 + 0.931i)T \)
97 \( 1 + (0.311 + 0.950i)T \)
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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

−18.00117648091499649369426159344, −17.57342882193908157138480900967, −16.84185786096706733024391291772, −16.47669137414479070561239307129, −15.464784677336905510005640268200, −14.79636276360292161818809530778, −13.75510204903815774622606681048, −13.02010968921542976206902915332, −12.60679542474343602516623130404, −11.765760731638649159770493219385, −11.20286581922754165001956441580, −10.75171627217907417198308490783, −9.82464121234632715402308415860, −9.41450193423085645123842637112, −8.62626978604749163316619489462, −7.65571034623818154343430226005, −7.03461444052267227496447940282, −6.27165724683011568689828072392, −5.72864217285350630450985957179, −4.92431748953862092645464085573, −3.72114013305207654040899937059, −2.8608264175285526810247326992, −2.244827415026833340580601626877, −1.501602484540875799419963924823, −0.45520744915753251313629521629, 0.78864120212511114573519344718, 1.335890073325308764544332606718, 2.3420463275064100317083326134, 3.72198899701554917987935089559, 4.74341492649125232698314970519, 5.06697108746130702529562113985, 5.752583150416980929681108728875, 6.7712235214526499570490112116, 7.08187762288126545445438193465, 7.74425886573186944734314836037, 9.17095698807951038301744502857, 9.40104729371843877864819961846, 9.955796604892987605834466627147, 10.54118048824589401163728847957, 11.39244104990929731178787525621, 12.09204182072652674097746890292, 12.93155118888384802572703171638, 13.608167038136661754028850745178, 14.542908679502442504745310205543, 14.947938312888897383745994195772, 15.99203127137669236313975392528, 16.63994187655535079747633396736, 16.74923161827382637449370351747, 17.45720867581721244834524573237, 18.06013693949562125394363471735

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