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Properties

Label 1-4729-4729.52-r0-0-0
Degree $1$
Conductor $4729$
Sign $-0.607 + 0.794i$
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  + (−0.848 − 0.529i)2-s + (0.783 − 0.620i)3-s + (0.439 + 0.898i)4-s + (0.341 − 0.939i)5-s + (−0.993 + 0.111i)6-s + (0.756 − 0.653i)7-s + (0.103 − 0.994i)8-s + (0.229 − 0.973i)9-s + (−0.787 + 0.616i)10-s + (−0.963 − 0.267i)11-s + (0.901 + 0.431i)12-s + (−0.859 + 0.511i)13-s + (−0.988 + 0.153i)14-s + (−0.316 − 0.948i)15-s + (−0.614 + 0.788i)16-s + (0.749 + 0.661i)17-s + ⋯
L(s)  = 1  + (−0.848 − 0.529i)2-s + (0.783 − 0.620i)3-s + (0.439 + 0.898i)4-s + (0.341 − 0.939i)5-s + (−0.993 + 0.111i)6-s + (0.756 − 0.653i)7-s + (0.103 − 0.994i)8-s + (0.229 − 0.973i)9-s + (−0.787 + 0.616i)10-s + (−0.963 − 0.267i)11-s + (0.901 + 0.431i)12-s + (−0.859 + 0.511i)13-s + (−0.988 + 0.153i)14-s + (−0.316 − 0.948i)15-s + (−0.614 + 0.788i)16-s + (0.749 + 0.661i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.607 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.607 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $-0.607 + 0.794i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (52, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ -0.607 + 0.794i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4742472821 - 0.9597980196i\)
\(L(\frac12)\) \(\approx\) \(-0.4742472821 - 0.9597980196i\)
\(L(1)\) \(\approx\) \(0.6192406964 - 0.6612896334i\)
\(L(1)\) \(\approx\) \(0.6192406964 - 0.6612896334i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (-0.848 - 0.529i)T \)
3 \( 1 + (0.783 - 0.620i)T \)
5 \( 1 + (0.341 - 0.939i)T \)
7 \( 1 + (0.756 - 0.653i)T \)
11 \( 1 + (-0.963 - 0.267i)T \)
13 \( 1 + (-0.859 + 0.511i)T \)
17 \( 1 + (0.749 + 0.661i)T \)
19 \( 1 + (0.156 - 0.987i)T \)
23 \( 1 + (-0.880 - 0.474i)T \)
29 \( 1 + (-0.848 + 0.529i)T \)
31 \( 1 + (-0.894 - 0.446i)T \)
37 \( 1 + (0.0928 - 0.995i)T \)
41 \( 1 + (0.839 + 0.543i)T \)
43 \( 1 + (0.923 + 0.383i)T \)
47 \( 1 + (-0.717 - 0.696i)T \)
53 \( 1 + (0.845 - 0.534i)T \)
59 \( 1 + (-0.702 - 0.711i)T \)
61 \( 1 + (-0.995 - 0.0902i)T \)
67 \( 1 + (0.985 + 0.169i)T \)
71 \( 1 + (0.972 + 0.231i)T \)
73 \( 1 + (0.659 + 0.751i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (0.877 - 0.479i)T \)
89 \( 1 + (0.531 + 0.846i)T \)
97 \( 1 + (0.756 - 0.653i)T \)
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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

−18.41317186751573864399410721820, −18.17086139603393263627421037853, −17.2923781698216219725645887222, −16.5746244071960285918568984764, −15.68007783564717650139797862060, −15.34730925445087641636385794715, −14.64800902808398114821777162093, −14.27771988299144329359769847698, −13.646313028693927022893568478691, −12.446665004595511245348440514519, −11.60955011144584536728547303049, −10.72149222313895335826789728494, −10.357231169006354074052880568333, −9.603569393229082964440539595996, −9.25562690119022991050914362971, −8.09209720476278996551232822835, −7.67805481588702674071517126158, −7.39970921670569662195917543813, −6.04218763560555649969808413885, −5.43421335890799746703699069659, −4.948107611209117185102698341090, −3.70267772642642222312124609567, −2.68678622036567678194734465319, −2.315527048294618264423952782493, −1.52435295949688496885067173736, 0.32572633755853375241869544854, 1.10967957308993290974594461084, 2.040827979184130639068278198101, 2.29414093080165229212160321558, 3.48449325275171046538508084253, 4.184165205260561859217741074038, 5.05068370866946875446111616715, 6.04417978791463735872476162540, 7.11386200012194923397161833310, 7.69847432693743742093195490779, 8.080590743768746506606926481757, 8.84416898747369101936612496517, 9.45917388043659325611293239715, 10.05718015030781729404660329728, 10.92491929412714201366495910586, 11.60362540745380388435088871334, 12.59784519949148474438027048648, 12.74388645365913876133477745640, 13.56207918178726782467430899117, 14.25423333509390552661939977905, 14.96748954472032747338412723069, 15.975379117942733568366633284855, 16.59796522167101109315084498824, 17.193391025451306471652889729937, 17.9095299486716341503893692632

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