↑

Properties

Label 1-4729-4729.304-r0-0-0
Degree $1$
Conductor $4729$
Sign $-0.934 + 0.355i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.981 − 0.190i)2-s + (−0.921 + 0.388i)3-s + (0.927 − 0.373i)4-s + (−0.0451 + 0.998i)5-s + (−0.830 + 0.556i)6-s + (0.749 + 0.661i)7-s + (0.839 − 0.543i)8-s + (0.698 − 0.715i)9-s + (0.145 + 0.989i)10-s + (−0.305 + 0.952i)11-s + (−0.709 + 0.704i)12-s + (0.361 + 0.932i)13-s + (0.862 + 0.506i)14-s + (−0.346 − 0.938i)15-s + (0.721 − 0.692i)16-s + (−0.976 + 0.216i)17-s + ⋯
L(s)  = 1  + (0.981 − 0.190i)2-s + (−0.921 + 0.388i)3-s + (0.927 − 0.373i)4-s + (−0.0451 + 0.998i)5-s + (−0.830 + 0.556i)6-s + (0.749 + 0.661i)7-s + (0.839 − 0.543i)8-s + (0.698 − 0.715i)9-s + (0.145 + 0.989i)10-s + (−0.305 + 0.952i)11-s + (−0.709 + 0.704i)12-s + (0.361 + 0.932i)13-s + (0.862 + 0.506i)14-s + (−0.346 − 0.938i)15-s + (0.721 − 0.692i)16-s + (−0.976 + 0.216i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.934 + 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.934 + 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $-0.934 + 0.355i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (304, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ -0.934 + 0.355i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3237103470 + 1.763052427i\)
\(L(\frac12)\) \(\approx\) \(0.3237103470 + 1.763052427i\)
\(L(1)\) \(\approx\) \(1.295590705 + 0.5517176468i\)
\(L(1)\) \(\approx\) \(1.295590705 + 0.5517176468i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (0.981 - 0.190i)T \)
3 \( 1 + (-0.921 + 0.388i)T \)
5 \( 1 + (-0.0451 + 0.998i)T \)
7 \( 1 + (0.749 + 0.661i)T \)
11 \( 1 + (-0.305 + 0.952i)T \)
13 \( 1 + (0.361 + 0.932i)T \)
17 \( 1 + (-0.976 + 0.216i)T \)
19 \( 1 + (0.370 - 0.928i)T \)
23 \( 1 + (-0.996 + 0.0796i)T \)
29 \( 1 + (0.981 + 0.190i)T \)
31 \( 1 + (-0.0239 - 0.999i)T \)
37 \( 1 + (-0.472 + 0.881i)T \)
41 \( 1 + (-0.536 + 0.843i)T \)
43 \( 1 + (-0.986 + 0.164i)T \)
47 \( 1 + (-0.885 - 0.465i)T \)
53 \( 1 + (0.0610 + 0.998i)T \)
59 \( 1 + (-0.0770 - 0.997i)T \)
61 \( 1 + (-0.913 + 0.407i)T \)
67 \( 1 + (-0.224 + 0.974i)T \)
71 \( 1 + (0.114 - 0.993i)T \)
73 \( 1 + (-0.904 + 0.427i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.171 - 0.985i)T \)
89 \( 1 + (0.659 + 0.751i)T \)
97 \( 1 + (0.749 + 0.661i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.78981139223525919413435551963, −17.0389646661432406258912980006, −16.418294830458806617189865087561, −15.943424973480334912621794959985, −15.42680576883912814291742036173, −14.1513039416266237624302169474, −13.77094787485424057151453056398, −13.16546946911949656466279291955, −12.50890327880907492142598193939, −11.90114246937897800067818064346, −11.31668441881965515808749677127, −10.60396953868264561616666833801, −10.117284002452497424321445539063, −8.49365664588515749959399434893, −8.18820684669067274715875058411, −7.4466685968686170133604211416, −6.61660157020732520485644521376, −5.802517088182823317833965143771, −5.36422543801535421328579586989, −4.707870681417346917984163688777, −4.05546707467252442073024116882, −3.21179987293638857540181602833, −1.92009591102075266358637118579, −1.356100529581564430450428036122, −0.344522604064986375403522388628, 1.50903423004329129635798989563, 2.09999784452151546691095437577, 2.9266321845961284650482372486, 3.905175222246762507338903896277, 4.67360586678237029711765072634, 4.93342393175887422425199947954, 6.07851796559402833201040190643, 6.44584498722180619967589182133, 7.09163582565091422083522979127, 7.9141130440978842516938444930, 9.104933439831606232484090893529, 9.99853882627785317322984069987, 10.4812391228582986336854758693, 11.32686450731711717335652739476, 11.65914061113376327818815079446, 12.07359643791275168213779451564, 13.144320271779882001495280053742, 13.70500307023388042356998110742, 14.630426694632969201647281911698, 15.10342336575057629769221100639, 15.599272878952139133125856794980, 16.13161798086035632467171160511, 17.17991050346231503367395201840, 17.84410369153998220058076329673, 18.340889218723273329935114657720

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