↑

Properties

Label 1-4729-4729.1018-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

Related objects

Downloads

Learn more

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: $\chi_{4729} (1018, \cdot )$
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 \( 1 + (0.311 - 0.950i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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