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Properties

Label 1-4729-4729.324-r0-0-0
Degree $1$
Conductor $4729$
Sign $-0.753 + 0.657i$
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  + (−0.996 − 0.0796i)2-s + (0.995 − 0.0955i)3-s + (0.987 + 0.158i)4-s + (0.198 − 0.980i)5-s + (−0.999 + 0.0159i)6-s + (−0.998 − 0.0478i)7-s + (−0.971 − 0.236i)8-s + (0.981 − 0.190i)9-s + (−0.275 + 0.961i)10-s + (−0.260 + 0.965i)11-s + (0.997 + 0.0637i)12-s + (−0.150 − 0.988i)13-s + (0.991 + 0.127i)14-s + (0.103 − 0.994i)15-s + (0.949 + 0.313i)16-s + (−0.763 − 0.645i)17-s + ⋯
L(s)  = 1  + (−0.996 − 0.0796i)2-s + (0.995 − 0.0955i)3-s + (0.987 + 0.158i)4-s + (0.198 − 0.980i)5-s + (−0.999 + 0.0159i)6-s + (−0.998 − 0.0478i)7-s + (−0.971 − 0.236i)8-s + (0.981 − 0.190i)9-s + (−0.275 + 0.961i)10-s + (−0.260 + 0.965i)11-s + (0.997 + 0.0637i)12-s + (−0.150 − 0.988i)13-s + (0.991 + 0.127i)14-s + (0.103 − 0.994i)15-s + (0.949 + 0.313i)16-s + (−0.763 − 0.645i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.753 + 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.753 + 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $-0.753 + 0.657i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (324, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ -0.753 + 0.657i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.07633737934 - 0.2035811038i\)
\(L(\frac12)\) \(\approx\) \(-0.07633737934 - 0.2035811038i\)
\(L(1)\) \(\approx\) \(0.7021847518 - 0.2290069300i\)
\(L(1)\) \(\approx\) \(0.7021847518 - 0.2290069300i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (-0.996 - 0.0796i)T \)
3 \( 1 + (0.995 - 0.0955i)T \)
5 \( 1 + (0.198 - 0.980i)T \)
7 \( 1 + (-0.998 - 0.0478i)T \)
11 \( 1 + (-0.260 + 0.965i)T \)
13 \( 1 + (-0.150 - 0.988i)T \)
17 \( 1 + (-0.763 - 0.645i)T \)
19 \( 1 + (-0.663 - 0.748i)T \)
23 \( 1 + (-0.290 + 0.956i)T \)
29 \( 1 + (0.996 - 0.0796i)T \)
31 \( 1 + (-0.927 + 0.373i)T \)
37 \( 1 + (0.872 - 0.488i)T \)
41 \( 1 + (0.933 - 0.358i)T \)
43 \( 1 + (-0.856 + 0.516i)T \)
47 \( 1 + (-0.803 + 0.595i)T \)
53 \( 1 + (-0.439 - 0.898i)T \)
59 \( 1 + (0.651 + 0.758i)T \)
61 \( 1 + (0.0875 - 0.996i)T \)
67 \( 1 + (0.839 - 0.543i)T \)
71 \( 1 + (-0.709 - 0.704i)T \)
73 \( 1 + (-0.963 - 0.267i)T \)
79 \( 1 - T \)
83 \( 1 + (0.783 - 0.620i)T \)
89 \( 1 + (-0.495 + 0.868i)T \)
97 \( 1 + (-0.998 - 0.0478i)T \)
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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

−18.58536358664055873535891995565, −18.32044535257279885608224249887, −17.20871092323824469743453078372, −16.40054918162498518424786927515, −16.07640262761471587508447639802, −15.23173719160653648365005427026, −14.63721613621518612357669642273, −14.11398131011086723029798015643, −13.234721114598346966492962151684, −12.60337774293447046283415120623, −11.56069136247996301062214016815, −10.871769560331140264821662218312, −10.12719138305207548023648820126, −9.86422847119304116068276059305, −8.855971193113100034270071128606, −8.5340208638742846991091250279, −7.69734881163049262831397340993, −6.83443915666291657060741187427, −6.45755437031433646705020130977, −5.82368667599415451965093072634, −4.24067953873639508445799807414, −3.56693639733100194777154614811, −2.71855646561710739509771638670, −2.35003999807172416721464326411, −1.43624219631356950979489336210, 0.069361034972540049871068432534, 1.06025466450929103486960928814, 2.02371335916256796520565289536, 2.594987135104717642848502070699, 3.35510645666908886940967518898, 4.31041851059844936540585267124, 5.16682376322291058060045682527, 6.23341769502271543071650081736, 6.94025770847236248901321302924, 7.63096145077709268361622173744, 8.207782907860335192089987148932, 9.0550847751826187527947211318, 9.42006353800236565609289377341, 9.94691748827875687358037857723, 10.64124663114192024654408505368, 11.70579159107121843976322903895, 12.61622499693636037633171018998, 12.90084815796596896293008469438, 13.44787904817824664166195861513, 14.60587560856661258311912780263, 15.30760118420736791647778835575, 15.96337459355366673926919454155, 16.1391641028838738403613483947, 17.32575050122363492175053910798, 17.778289572498979348435838069892

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