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Properties

Label 1-4729-4729.8-r0-0-0
Degree $1$
Conductor $4729$
Sign $0.721 - 0.691i$
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 + (−0.522 − 0.852i)5-s + (−0.978 + 0.205i)6-s + (0.812 + 0.582i)7-s + (−0.999 + 0.0318i)8-s + (−0.793 − 0.608i)9-s + (0.467 − 0.884i)10-s + (−0.961 − 0.275i)11-s + (−0.675 − 0.737i)12-s + (−0.921 + 0.388i)13-s + (−0.0875 + 0.996i)14-s + (0.975 − 0.221i)15-s + (−0.536 − 0.843i)16-s + (0.298 + 0.954i)17-s + ⋯
L(s)  = 1  + (0.509 + 0.860i)2-s + (−0.321 + 0.947i)3-s + (−0.481 + 0.876i)4-s + (−0.522 − 0.852i)5-s + (−0.978 + 0.205i)6-s + (0.812 + 0.582i)7-s + (−0.999 + 0.0318i)8-s + (−0.793 − 0.608i)9-s + (0.467 − 0.884i)10-s + (−0.961 − 0.275i)11-s + (−0.675 − 0.737i)12-s + (−0.921 + 0.388i)13-s + (−0.0875 + 0.996i)14-s + (0.975 − 0.221i)15-s + (−0.536 − 0.843i)16-s + (0.298 + 0.954i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.721 - 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.721 - 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $0.721 - 0.691i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ 0.721 - 0.691i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2813330461 + 0.1130487564i\)
\(L(\frac12)\) \(\approx\) \(-0.2813330461 + 0.1130487564i\)
\(L(1)\) \(\approx\) \(0.5302049992 + 0.6603448868i\)
\(L(1)\) \(\approx\) \(0.5302049992 + 0.6603448868i\)

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 \( 1 + (-0.522 - 0.852i)T \)
7 \( 1 + (0.812 + 0.582i)T \)
11 \( 1 + (-0.961 - 0.275i)T \)
13 \( 1 + (-0.921 + 0.388i)T \)
17 \( 1 + (0.298 + 0.954i)T \)
19 \( 1 + (0.00797 + 0.999i)T \)
23 \( 1 + (0.768 + 0.639i)T \)
29 \( 1 + (0.860 + 0.509i)T \)
31 \( 1 + (-0.965 + 0.260i)T \)
37 \( 1 + (0.939 - 0.343i)T \)
41 \( 1 + (-0.998 + 0.0557i)T \)
43 \( 1 + (-0.692 - 0.721i)T \)
47 \( 1 + (-0.905 + 0.424i)T \)
53 \( 1 + (0.930 + 0.366i)T \)
59 \( 1 + (-0.198 + 0.980i)T \)
61 \( 1 + (0.417 + 0.908i)T \)
67 \( 1 + (0.380 - 0.924i)T \)
71 \( 1 + (-0.742 - 0.669i)T \)
73 \( 1 + (-0.927 - 0.373i)T \)
79 \( 1 + iT \)
83 \( 1 + (-0.753 - 0.657i)T \)
89 \( 1 + (-0.898 - 0.439i)T \)
97 \( 1 + (-0.812 - 0.582i)T \)
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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

−17.900857161207552827404575708849, −17.23212119578448598098905627314, −16.21918663316826571763064264483, −15.19488646223675814032193576296, −14.71763002648806463393633117983, −14.13789948483507883991463138839, −13.34358786660778891059132866713, −12.93602700057015226742635989225, −12.04246886628926205407532475809, −11.450253988417256003145159990636, −11.09682921955951354435561538642, −10.32240044997832607619676930830, −9.7635862261195980971887213047, −8.450142682679830584907214073030, −7.88253725230206556950115718440, −7.045468541873613292417277997013, −6.72693888070700547004428922832, −5.43746883886099585129695269639, −4.98501209795538140175851770427, −4.32275063595146515741128797023, −2.996152312827909527844140367070, −2.72200095801257710982722232781, −1.93584320536439064847798843689, −0.82226500605335091110615362111, −0.08861641270737717296504487234, 1.47065047025546920043308940144, 2.74994246736945838993253014458, 3.55414003830777398658753528565, 4.3213402130102030309955375289, 4.9163110838069763150571647539, 5.44510454988454786160254196144, 5.868980711309962195982605103990, 7.105355299691875369582520319023, 7.885641543037828410602499148703, 8.45390018479896027688755060096, 8.94217899578993415085889773477, 9.786823439996236376107638191300, 10.652717462661819660551635260990, 11.531213889542662225410701040739, 12.10106261286458499089466437623, 12.59041557784476744867761309497, 13.43513123924464530715017417683, 14.39324642226599444905102328271, 14.96475919764474221712795198090, 15.28332790269915350714500450663, 16.13401593726265808751544589349, 16.59805192186681203436076768034, 17.0694295823222544655266550020, 17.82999574233832890334540527423, 18.49787949417545896079342877120

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