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Properties

Label 1-4729-4729.670-r0-0-0
Degree $1$
Conductor $4729$
Sign $0.146 - 0.989i$
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 + (−0.978 − 0.205i)5-s + (0.549 + 0.835i)6-s + (−0.984 − 0.174i)7-s + (−0.639 − 0.768i)8-s + (0.763 − 0.645i)9-s + (−0.0239 − 0.999i)10-s + (−0.821 + 0.569i)11-s + (−0.687 + 0.726i)12-s + (0.999 − 0.0318i)13-s + (−0.0557 − 0.998i)14-s + (−0.989 + 0.143i)15-s + (0.601 − 0.798i)16-s + (−0.887 − 0.460i)17-s + ⋯
L(s)  = 1  + (0.229 + 0.973i)2-s + (0.939 − 0.343i)3-s + (−0.894 + 0.446i)4-s + (−0.978 − 0.205i)5-s + (0.549 + 0.835i)6-s + (−0.984 − 0.174i)7-s + (−0.639 − 0.768i)8-s + (0.763 − 0.645i)9-s + (−0.0239 − 0.999i)10-s + (−0.821 + 0.569i)11-s + (−0.687 + 0.726i)12-s + (0.999 − 0.0318i)13-s + (−0.0557 − 0.998i)14-s + (−0.989 + 0.143i)15-s + (0.601 − 0.798i)16-s + (−0.887 − 0.460i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.146 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.146 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $0.146 - 0.989i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (670, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ 0.146 - 0.989i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4894459135 - 0.4221205654i\)
\(L(\frac12)\) \(\approx\) \(0.4894459135 - 0.4221205654i\)
\(L(1)\) \(\approx\) \(0.9155596052 + 0.2610432525i\)
\(L(1)\) \(\approx\) \(0.9155596052 + 0.2610432525i\)

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 \( 1 + (-0.978 - 0.205i)T \)
7 \( 1 + (-0.984 - 0.174i)T \)
11 \( 1 + (-0.821 + 0.569i)T \)
13 \( 1 + (0.999 - 0.0318i)T \)
17 \( 1 + (-0.887 - 0.460i)T \)
19 \( 1 + (-0.536 - 0.843i)T \)
23 \( 1 + (0.848 + 0.529i)T \)
29 \( 1 + (-0.229 + 0.973i)T \)
31 \( 1 + (-0.166 + 0.986i)T \)
37 \( 1 + (0.975 + 0.221i)T \)
41 \( 1 + (0.732 + 0.681i)T \)
43 \( 1 + (0.589 - 0.808i)T \)
47 \( 1 + (0.275 - 0.961i)T \)
53 \( 1 + (-0.995 + 0.0955i)T \)
59 \( 1 + (-0.999 - 0.0159i)T \)
61 \( 1 + (0.663 + 0.748i)T \)
67 \( 1 + (-0.509 - 0.860i)T \)
71 \( 1 + (-0.244 + 0.969i)T \)
73 \( 1 + (-0.453 + 0.891i)T \)
79 \( 1 - T \)
83 \( 1 + (-0.773 - 0.633i)T \)
89 \( 1 + (0.753 - 0.657i)T \)
97 \( 1 + (-0.984 - 0.174i)T \)
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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

−18.77713625787929930218922217179, −18.06832069273528685771770114983, −16.818769761182621691471875653392, −16.05016465714574810909535238090, −15.532477875591093078724639835502, −14.95777985908817449942398070866, −14.23476348522901552806129854617, −13.376505730976247338416087450037, −12.94613200594126927730517429465, −12.50187269944101949598848841366, −11.300325505059175600425283101973, −10.88637669033391698596398243513, −10.335666395389383609666379110256, −9.39829019623943355780800156508, −8.90257969574423839998173644154, −8.17103325697733395112652836315, −7.67902006460068606846273969657, −6.38233390180694955039682572843, −5.82282642698563613031336882285, −4.52420340528330662728371557823, −4.08782859512988935304680913856, −3.451075039819205567282715407174, −2.79215917830379985135800971962, −2.24577664696698119203040080633, −0.98056952453456751114497414340, 0.169412629973874108986976918132, 1.255758709407122637362591954170, 2.79549617931582254579858073454, 3.14048052039667411812635342845, 4.08588513831070439196645091842, 4.53413700183380174535292898442, 5.52003005515566788357986036045, 6.61129560848846899373645758518, 7.04735076113472532431289265862, 7.52241884695432728759695205791, 8.40957471943044524323766438629, 8.92053708205605112057436354159, 9.39043397163288843252333316671, 10.43532350331824855375263860744, 11.28302197618589533380969255551, 12.40827740590388023286207155403, 12.94194440154396985538690548293, 13.22209018033522687321364361390, 13.95242855367384022679605373972, 14.92105710126902762926956563868, 15.35571822828738430662509346367, 15.92231688583975505996058047460, 16.23611068310477124491797808245, 17.32720536937141352035604104641, 18.07915332611086138192893806226

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