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Properties

Label 1-4729-4729.2301-r0-0-0
Degree $1$
Conductor $4729$
Sign $-0.386 + 0.922i$
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  + (−0.536 + 0.843i)2-s + (0.260 − 0.965i)3-s + (−0.424 − 0.905i)4-s + (−0.589 + 0.808i)5-s + (0.675 + 0.737i)6-s + (−0.793 − 0.608i)7-s + (0.991 + 0.127i)8-s + (−0.864 − 0.502i)9-s + (−0.366 − 0.930i)10-s + (−0.439 + 0.898i)11-s + (−0.984 + 0.174i)12-s + (−0.0239 + 0.999i)13-s + (0.939 − 0.343i)14-s + (0.627 + 0.778i)15-s + (−0.639 + 0.768i)16-s + (−0.351 − 0.936i)17-s + ⋯
L(s)  = 1  + (−0.536 + 0.843i)2-s + (0.260 − 0.965i)3-s + (−0.424 − 0.905i)4-s + (−0.589 + 0.808i)5-s + (0.675 + 0.737i)6-s + (−0.793 − 0.608i)7-s + (0.991 + 0.127i)8-s + (−0.864 − 0.502i)9-s + (−0.366 − 0.930i)10-s + (−0.439 + 0.898i)11-s + (−0.984 + 0.174i)12-s + (−0.0239 + 0.999i)13-s + (0.939 − 0.343i)14-s + (0.627 + 0.778i)15-s + (−0.639 + 0.768i)16-s + (−0.351 − 0.936i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $-0.386 + 0.922i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (2301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ -0.386 + 0.922i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2973808815 + 0.4472729745i\)
\(L(\frac12)\) \(\approx\) \(0.2973808815 + 0.4472729745i\)
\(L(1)\) \(\approx\) \(0.6108537453 + 0.1196681504i\)
\(L(1)\) \(\approx\) \(0.6108537453 + 0.1196681504i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (-0.536 + 0.843i)T \)
3 \( 1 + (0.260 - 0.965i)T \)
5 \( 1 + (-0.589 + 0.808i)T \)
7 \( 1 + (-0.793 - 0.608i)T \)
11 \( 1 + (-0.439 + 0.898i)T \)
13 \( 1 + (-0.0239 + 0.999i)T \)
17 \( 1 + (-0.351 - 0.936i)T \)
19 \( 1 + (0.999 + 0.0318i)T \)
23 \( 1 + (0.933 + 0.358i)T \)
29 \( 1 + (0.536 + 0.843i)T \)
31 \( 1 + (-0.495 - 0.868i)T \)
37 \( 1 + (0.166 + 0.986i)T \)
41 \( 1 + (-0.975 - 0.221i)T \)
43 \( 1 + (0.996 - 0.0796i)T \)
47 \( 1 + (0.182 - 0.983i)T \)
53 \( 1 + (-0.0717 + 0.997i)T \)
59 \( 1 + (0.698 - 0.715i)T \)
61 \( 1 + (0.150 - 0.988i)T \)
67 \( 1 + (0.00797 - 0.999i)T \)
71 \( 1 + (-0.978 - 0.205i)T \)
73 \( 1 + (0.0398 - 0.999i)T \)
79 \( 1 - T \)
83 \( 1 + (-0.963 - 0.267i)T \)
89 \( 1 + (0.244 + 0.969i)T \)
97 \( 1 + (-0.793 - 0.608i)T \)
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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

−17.91616495453558733050454562920, −17.29124918802143786587467067284, −16.476154971236493110262012946854, −16.003902670187046946753687950613, −15.616719436904940872498826022815, −14.7337605929633359782752654401, −13.70759635934804344969737204777, −12.946732319317831826350523782827, −12.64897439629406949704555281611, −11.65111686705680433176098442580, −11.17270038816044036220778015971, −10.347491559498521785678450602326, −9.88433769435806093268329345171, −8.910576622427865396564244523554, −8.70611233260153596502959710418, −8.07714098668110624625985270414, −7.226018152184323500076622809742, −5.78629181331350372924026054208, −5.33814272899272185834522936837, −4.40161162254700890707050926678, −3.693413160473559985898580274709, −3.03660454112171779894021880603, −2.58050291198525603470213127723, −1.18764577368022651014687700950, −0.24848464098475546128053441585, 0.76671840588360170252481823906, 1.77616390984281103218258550019, 2.70567334033022834867082474530, 3.48109529257255252472312643607, 4.47817664112451113281822921435, 5.31083544536622820260605145889, 6.406981220147927118693629720591, 6.83554547497247658035588913851, 7.36704918230949224983196836733, 7.641287985523087512333535689356, 8.74105712204327024857111623058, 9.408118026244337297863312872431, 9.99518414943814900653642114303, 10.90361450521749646760457449826, 11.56830255518583598681241496022, 12.34192831962117446283584894618, 13.29557899786438925395366970155, 13.770981057006565236412224088912, 14.33418108586081985028039489816, 15.10027392673904293084138902921, 15.68066649735677223064506667523, 16.381264734213931688870337873387, 17.09522034430126310341238176331, 17.78597455960700839391506354746, 18.545188176147826577261403807910

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