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Properties

Label 1-4729-4729.1837-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: $\chi_{4729} (1837, \cdot )$
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

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

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