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Properties

Label 1-4729-4729.1034-r0-0-0
Degree $1$
Conductor $4729$
Sign $-0.949 - 0.312i$
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  + (−0.229 − 0.973i)2-s + (−0.939 + 0.343i)3-s + (−0.894 + 0.446i)4-s + (−0.667 + 0.744i)5-s + (0.549 + 0.835i)6-s + (−0.643 + 0.765i)7-s + (0.639 + 0.768i)8-s + (0.763 − 0.645i)9-s + (0.877 + 0.479i)10-s + (0.569 + 0.821i)11-s + (0.687 − 0.726i)12-s + (−0.472 + 0.881i)13-s + (0.892 + 0.450i)14-s + (0.370 − 0.928i)15-s + (0.601 − 0.798i)16-s + (0.538 + 0.842i)17-s + ⋯
L(s)  = 1  + (−0.229 − 0.973i)2-s + (−0.939 + 0.343i)3-s + (−0.894 + 0.446i)4-s + (−0.667 + 0.744i)5-s + (0.549 + 0.835i)6-s + (−0.643 + 0.765i)7-s + (0.639 + 0.768i)8-s + (0.763 − 0.645i)9-s + (0.877 + 0.479i)10-s + (0.569 + 0.821i)11-s + (0.687 − 0.726i)12-s + (−0.472 + 0.881i)13-s + (0.892 + 0.450i)14-s + (0.370 − 0.928i)15-s + (0.601 − 0.798i)16-s + (0.538 + 0.842i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.949 - 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.949 - 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $-0.949 - 0.312i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (1034, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ -0.949 - 0.312i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.05947560289 + 0.3710815967i\)
\(L(\frac12)\) \(\approx\) \(-0.05947560289 + 0.3710815967i\)
\(L(1)\) \(\approx\) \(0.4935665818 + 0.1079877549i\)
\(L(1)\) \(\approx\) \(0.4935665818 + 0.1079877549i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (-0.229 - 0.973i)T \)
3 \( 1 + (-0.939 + 0.343i)T \)
5 \( 1 + (-0.667 + 0.744i)T \)
7 \( 1 + (-0.643 + 0.765i)T \)
11 \( 1 + (0.569 + 0.821i)T \)
13 \( 1 + (-0.472 + 0.881i)T \)
17 \( 1 + (0.538 + 0.842i)T \)
19 \( 1 + (0.999 - 0.0425i)T \)
23 \( 1 + (-0.529 + 0.848i)T \)
29 \( 1 + (-0.973 - 0.229i)T \)
31 \( 1 + (0.986 + 0.166i)T \)
37 \( 1 + (-0.679 + 0.733i)T \)
41 \( 1 + (-0.681 + 0.732i)T \)
43 \( 1 + (0.106 - 0.994i)T \)
47 \( 1 + (-0.719 + 0.694i)T \)
53 \( 1 + (-0.814 + 0.580i)T \)
59 \( 1 + (-0.486 - 0.873i)T \)
61 \( 1 + (0.948 + 0.316i)T \)
67 \( 1 + (-0.490 + 0.871i)T \)
71 \( 1 + (-0.962 + 0.272i)T \)
73 \( 1 + (0.545 + 0.838i)T \)
79 \( 1 + (0.866 - 0.5i)T \)
83 \( 1 + (-0.161 + 0.986i)T \)
89 \( 1 + (-0.981 + 0.192i)T \)
97 \( 1 + (0.643 - 0.765i)T \)
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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

−17.59892170795888542147800395239, −16.89921150730778452229664932255, −16.47340441997835411587835899900, −16.09053354220173861595533313769, −15.46662471706220563246513246180, −14.434178503295586850769188712876, −13.70919547799671757559117720112, −13.16134239967004774279349962133, −12.43636078489318615761928681521, −11.8500946790054788187489263132, −10.9901436205677862881946082090, −10.17173731239805850688206161085, −9.62157019311090866701852106095, −8.7772125211034858898119460276, −7.80980464036002561205792557683, −7.547369731443686688622948838153, −6.745697479195349573565084251677, −6.05295924668129201687463308313, −5.29618495049439397298607363208, −4.80926604454918286016786751361, −3.90243977638553203266891933755, −3.22164067622353025513326951625, −1.45674415669699508977884813372, −0.642306906671082898123991522590, −0.22556470422161756296616407783, 1.285336208479139986043670675319, 2.03522366718467551129997704339, 3.121919541266446528104017107407, 3.68101648232151507704355043504, 4.3667038932810604374437287489, 5.15148614508911745878067181115, 6.01990341951808221979582820862, 6.819251320452288868214133468181, 7.47087603316549274774150407382, 8.40003663396645265872807973639, 9.45521460749988450896039444196, 9.77452442105669011242285960405, 10.31963407300437435783255060240, 11.35492826670327945870811006823, 11.71502851955022543221280101850, 12.161535926137108636634346306596, 12.72053818153674996529825280078, 13.745500625719674794180738288070, 14.57265964616683400718467231152, 15.24680278365962840639540284916, 15.87869484050176788780181381926, 16.682494569936636796704325746887, 17.33947140593068520585163861681, 17.91174444188935637920994115173, 18.746219338069629061697409888245

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