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Properties

Label 1-4729-4729.1054-r0-0-0
Degree $1$
Conductor $4729$
Sign $-0.146 - 0.989i$
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  + (0.933 − 0.358i)2-s + (0.481 + 0.876i)3-s + (0.742 − 0.669i)4-s + (−0.923 + 0.383i)5-s + (0.763 + 0.645i)6-s + (−0.999 + 0.0106i)7-s + (0.453 − 0.891i)8-s + (−0.536 + 0.843i)9-s + (−0.724 + 0.689i)10-s + (−0.835 + 0.549i)11-s + (0.944 + 0.328i)12-s + (−0.787 + 0.616i)13-s + (−0.929 + 0.368i)14-s + (−0.780 − 0.624i)15-s + (0.103 − 0.994i)16-s + (0.483 + 0.875i)17-s + ⋯
L(s)  = 1  + (0.933 − 0.358i)2-s + (0.481 + 0.876i)3-s + (0.742 − 0.669i)4-s + (−0.923 + 0.383i)5-s + (0.763 + 0.645i)6-s + (−0.999 + 0.0106i)7-s + (0.453 − 0.891i)8-s + (−0.536 + 0.843i)9-s + (−0.724 + 0.689i)10-s + (−0.835 + 0.549i)11-s + (0.944 + 0.328i)12-s + (−0.787 + 0.616i)13-s + (−0.929 + 0.368i)14-s + (−0.780 − 0.624i)15-s + (0.103 − 0.994i)16-s + (0.483 + 0.875i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.146 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.146 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $-0.146 - 0.989i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (1054, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ -0.146 - 0.989i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3157204493 - 0.3659476545i\)
\(L(\frac12)\) \(\approx\) \(0.3157204493 - 0.3659476545i\)
\(L(1)\) \(\approx\) \(1.201685208 + 0.2071432772i\)
\(L(1)\) \(\approx\) \(1.201685208 + 0.2071432772i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (0.933 - 0.358i)T \)
3 \( 1 + (0.481 + 0.876i)T \)
5 \( 1 + (-0.923 + 0.383i)T \)
7 \( 1 + (-0.999 + 0.0106i)T \)
11 \( 1 + (-0.835 + 0.549i)T \)
13 \( 1 + (-0.787 + 0.616i)T \)
17 \( 1 + (0.483 + 0.875i)T \)
19 \( 1 + (-0.859 + 0.511i)T \)
23 \( 1 + (-0.402 - 0.915i)T \)
29 \( 1 + (-0.358 + 0.933i)T \)
31 \( 1 + (0.905 - 0.424i)T \)
37 \( 1 + (0.833 - 0.551i)T \)
41 \( 1 + (-0.417 + 0.908i)T \)
43 \( 1 + (0.226 + 0.973i)T \)
47 \( 1 + (-0.950 - 0.311i)T \)
53 \( 1 + (-0.811 - 0.584i)T \)
59 \( 1 + (0.655 + 0.755i)T \)
61 \( 1 + (0.323 - 0.946i)T \)
67 \( 1 + (-0.385 - 0.922i)T \)
71 \( 1 + (-0.341 - 0.939i)T \)
73 \( 1 + (0.917 - 0.397i)T \)
79 \( 1 + (0.866 - 0.5i)T \)
83 \( 1 + (-0.622 + 0.782i)T \)
89 \( 1 + (-0.958 + 0.285i)T \)
97 \( 1 + (0.999 - 0.0106i)T \)
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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

−18.55639030606928773803734462957, −17.43164727657407683978376521975, −16.9864337319753686258546583836, −15.983724799594260087890140908149, −15.66172626452265420151268140620, −15.04302549237379862413294759427, −14.24338106028065409878897642513, −13.34186834863365540743837529517, −13.18857426601747225159747752188, −12.40876167846971383668010530497, −11.917861389948417426663350522296, −11.28830703879963765051515393317, −10.24286076030943108626183777860, −9.328916422550484854980632324691, −8.357707271342259316263350859506, −7.9302236166889786570519380129, −7.26469040136840871096553234435, −6.73366711627982058312745938158, −5.80786594634299927284315406312, −5.234981181270760325695016876589, −4.2470722200625793268078597478, −3.44413140779083759591741041139, −2.8651409402011168137000742465, −2.33877236433790792486316402629, −0.88842786211712322876592470249, 0.09355775712299197153837347174, 1.874821996885263360690593526582, 2.647547195557610428899621277083, 3.18119683598502538779975359020, 3.96250609449469305060649976947, 4.4411028986188051711053208743, 5.122158053789750958542703949291, 6.192785640923460169178788490753, 6.71698877330338421355603835838, 7.73138678879824547425590077131, 8.21699472677925262106294860730, 9.435923330026936281047863534136, 9.99742905863100334241644882145, 10.53275623978596809135142474901, 11.12037018000131533576196956686, 12.05798453763056481908824581822, 12.6225802458290928756852938542, 13.13958632114787572682857812519, 14.16745718586152436907786959122, 14.865956164412107657379114326017, 14.96622133637402070400214138642, 15.837256671925362279785753291258, 16.438783715402146849165554847683, 16.77135544243261525156751272583, 18.29889751229179410933866518582

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