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Properties

Label 1-4729-4729.1022-r0-0-0
Degree $1$
Conductor $4729$
Sign $-0.148 - 0.988i$
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 + (0.129 − 0.991i)5-s + (0.763 + 0.645i)6-s + (0.490 − 0.871i)7-s + (0.453 − 0.891i)8-s + (−0.536 + 0.843i)9-s + (−0.234 − 0.972i)10-s + (0.835 − 0.549i)11-s + (0.944 + 0.328i)12-s + (−0.140 − 0.990i)13-s + (0.145 − 0.989i)14-s + (0.931 − 0.363i)15-s + (0.103 − 0.994i)16-s + (0.999 + 0.0186i)17-s + ⋯
L(s)  = 1  + (0.933 − 0.358i)2-s + (0.481 + 0.876i)3-s + (0.742 − 0.669i)4-s + (0.129 − 0.991i)5-s + (0.763 + 0.645i)6-s + (0.490 − 0.871i)7-s + (0.453 − 0.891i)8-s + (−0.536 + 0.843i)9-s + (−0.234 − 0.972i)10-s + (0.835 − 0.549i)11-s + (0.944 + 0.328i)12-s + (−0.140 − 0.990i)13-s + (0.145 − 0.989i)14-s + (0.931 − 0.363i)15-s + (0.103 − 0.994i)16-s + (0.999 + 0.0186i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.148 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.148 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $-0.148 - 0.988i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (1022, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ -0.148 - 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.080993321 - 3.579884233i\)
\(L(\frac12)\) \(\approx\) \(3.080993321 - 3.579884233i\)
\(L(1)\) \(\approx\) \(2.245608188 - 0.9691640194i\)
\(L(1)\) \(\approx\) \(2.245608188 - 0.9691640194i\)

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 \( 1 + (0.129 - 0.991i)T \)
7 \( 1 + (0.490 - 0.871i)T \)
11 \( 1 + (0.835 - 0.549i)T \)
13 \( 1 + (-0.140 - 0.990i)T \)
17 \( 1 + (0.999 + 0.0186i)T \)
19 \( 1 + (-0.0132 - 0.999i)T \)
23 \( 1 + (0.402 + 0.915i)T \)
29 \( 1 + (0.358 - 0.933i)T \)
31 \( 1 + (-0.905 + 0.424i)T \)
37 \( 1 + (0.0610 + 0.998i)T \)
41 \( 1 + (0.417 - 0.908i)T \)
43 \( 1 + (-0.730 + 0.683i)T \)
47 \( 1 + (-0.744 + 0.667i)T \)
53 \( 1 + (0.100 - 0.994i)T \)
59 \( 1 + (0.326 - 0.945i)T \)
61 \( 1 + (0.981 - 0.192i)T \)
67 \( 1 + (-0.606 + 0.795i)T \)
71 \( 1 + (-0.643 + 0.765i)T \)
73 \( 1 + (-0.114 + 0.993i)T \)
79 \( 1 + (0.866 + 0.5i)T \)
83 \( 1 + (0.988 + 0.148i)T \)
89 \( 1 + (-0.231 + 0.972i)T \)
97 \( 1 + (-0.490 + 0.871i)T \)
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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

−18.19178511658403693540083246466, −17.95937854489077523767049431646, −16.79688466082791170669831115671, −16.45073557749437417709606455855, −15.133614268594599972818895751898, −14.771975779409136333127637546825, −14.38778946537984388854438107329, −13.959387796440870664813241477649, −12.95919168916016707730490164690, −12.227201172975024730378962417422, −11.9211718515218400443233978682, −11.26345753955590274506228277413, −10.316221443935870689677594787279, −9.255621992580864817876827254605, −8.64505484745770015527986910067, −7.69638463364773615078972726734, −7.28257273331969103503742899957, −6.501036871251169220621965917450, −6.07495757024391071871330311707, −5.26500988608440678239819049356, −4.226639498022629894216596174247, −3.464644806985911032812802751387, −2.80363823259807162459576007135, −1.91832783848993049237908460752, −1.63217811976102892894750295766, 0.79449159519108975310913803326, 1.4180716080160267034452153921, 2.497596881362628499628013261231, 3.488722018641129583541073440628, 3.77875133380976331439897907119, 4.73474823543425807307530152230, 5.148256821172925656469287439129, 5.77154434739538243287819882496, 6.84796692285556999029992926648, 7.804483726350539428496383131647, 8.33891145088643773887284160914, 9.35836690324135033079909538059, 9.85375062711228100346171763735, 10.5394241472802479450815060227, 11.37785395161819191829415413919, 11.71114001799475688126693645309, 12.86671325638104858701907781857, 13.30462082909305716834214268193, 13.96067291966988542671968183799, 14.53212819726024453819029361179, 15.15612610992233708304806457837, 15.936207302742519274427544943018, 16.43578316944851201685605762340, 17.153563327390158203805353572608, 17.66013766551700918138090364836

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