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Properties

Label 1-4729-4729.1084-r0-0-0
Degree $1$
Conductor $4729$
Sign $-0.987 + 0.158i$
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  + (0.453 − 0.891i)2-s + (−0.997 − 0.0637i)3-s + (−0.589 − 0.808i)4-s + (−0.610 − 0.792i)5-s + (−0.509 + 0.860i)6-s + (0.472 − 0.881i)7-s + (−0.987 + 0.158i)8-s + (0.991 + 0.127i)9-s + (−0.982 + 0.184i)10-s + (0.174 + 0.984i)11-s + (0.536 + 0.843i)12-s + (−0.994 − 0.100i)13-s + (−0.571 − 0.820i)14-s + (0.558 + 0.829i)15-s + (−0.305 + 0.952i)16-s + (0.547 − 0.836i)17-s + ⋯
L(s)  = 1  + (0.453 − 0.891i)2-s + (−0.997 − 0.0637i)3-s + (−0.589 − 0.808i)4-s + (−0.610 − 0.792i)5-s + (−0.509 + 0.860i)6-s + (0.472 − 0.881i)7-s + (−0.987 + 0.158i)8-s + (0.991 + 0.127i)9-s + (−0.982 + 0.184i)10-s + (0.174 + 0.984i)11-s + (0.536 + 0.843i)12-s + (−0.994 − 0.100i)13-s + (−0.571 − 0.820i)14-s + (0.558 + 0.829i)15-s + (−0.305 + 0.952i)16-s + (0.547 − 0.836i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.158i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.158i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $-0.987 + 0.158i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (1084, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ -0.987 + 0.158i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.09396436252 - 1.177943508i\)
\(L(\frac12)\) \(\approx\) \(-0.09396436252 - 1.177943508i\)
\(L(1)\) \(\approx\) \(0.5735382933 - 0.6480564397i\)
\(L(1)\) \(\approx\) \(0.5735382933 - 0.6480564397i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (0.453 - 0.891i)T \)
3 \( 1 + (-0.997 - 0.0637i)T \)
5 \( 1 + (-0.610 - 0.792i)T \)
7 \( 1 + (0.472 - 0.881i)T \)
11 \( 1 + (0.174 + 0.984i)T \)
13 \( 1 + (-0.994 - 0.100i)T \)
17 \( 1 + (0.547 - 0.836i)T \)
19 \( 1 + (-0.885 - 0.465i)T \)
23 \( 1 + (0.947 + 0.321i)T \)
29 \( 1 + (0.891 - 0.453i)T \)
31 \( 1 + (0.252 - 0.967i)T \)
37 \( 1 + (0.942 + 0.333i)T \)
41 \( 1 + (0.961 - 0.275i)T \)
43 \( 1 + (-0.353 + 0.935i)T \)
47 \( 1 + (0.995 - 0.0981i)T \)
53 \( 1 + (-0.975 + 0.218i)T \)
59 \( 1 + (0.890 + 0.455i)T \)
61 \( 1 + (0.893 + 0.448i)T \)
67 \( 1 + (-0.140 - 0.990i)T \)
71 \( 1 + (0.00265 - 0.999i)T \)
73 \( 1 + (0.647 + 0.762i)T \)
79 \( 1 + (-0.866 + 0.5i)T \)
83 \( 1 + (-0.0770 - 0.997i)T \)
89 \( 1 + (-0.338 - 0.940i)T \)
97 \( 1 + (-0.472 + 0.881i)T \)
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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

−18.42318859592627287992821620321, −17.61985875983303481439039626516, −17.14937150557332805397601750820, −16.41871755716916064259395588721, −15.84302984903111484867212725476, −15.18747894236118771750140903139, −14.53605768222618414744686050306, −14.25156610753281944708289465536, −12.92448102809689243900971828006, −12.42820622561034470087258910615, −11.91106943936063284471899740087, −11.17732636065997427139484710884, −10.57598038587655366045210545482, −9.66961870907733238228435877536, −8.61271515095569177776884222843, −8.17645133413833154761716468356, −7.28589771358915791042303663510, −6.65409820247430577709882893116, −6.04691275072125897336760939154, −5.43468744009743890270027428174, −4.69283597387311817447188809456, −3.99982547495493653127587867251, −3.154380089983902076171893597165, −2.3143594489953789954271508912, −0.85717417419861292929593463973, 0.48915433700543808189891392164, 0.99549483553456865606400706604, 1.917184534745424182939983777362, 2.87799170474882307806758009524, 4.15958158694094256760357461534, 4.47720316569069046041732623444, 4.90112828123122085288716847875, 5.67768396646176092154005956323, 6.75622410286516245467125204706, 7.39729939560777572713934712763, 8.10352483803749362039073514041, 9.34976307485400280926169645623, 9.74937788219476686537539986669, 10.509965245802385306572361007322, 11.26476386039968574585513926986, 11.71709903182678992420348157919, 12.32889800784476764382516906389, 12.923954060976137927999252280351, 13.42042984126341121893974790097, 14.46237767689742047457714701591, 15.064386077425871129928310814245, 15.74015019916184263758417414043, 16.704615013301922793310671748644, 17.27423790618366571868056466983, 17.619251994109993629948155414894

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