↑

Properties

Label 1-4729-4729.381-r0-0-0
Degree $1$
Conductor $4729$
Sign $-0.936 - 0.351i$
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  + (0.522 + 0.852i)2-s + (0.999 + 0.0318i)3-s + (−0.453 + 0.891i)4-s + (−0.830 + 0.556i)5-s + (0.495 + 0.868i)6-s + (−0.999 + 0.0159i)7-s + (−0.996 + 0.0796i)8-s + (0.997 + 0.0637i)9-s + (−0.908 − 0.417i)10-s + (0.0875 + 0.996i)11-s + (−0.481 + 0.876i)12-s + (0.839 + 0.543i)13-s + (−0.536 − 0.843i)14-s + (−0.848 + 0.529i)15-s + (−0.589 − 0.808i)16-s + (0.687 + 0.726i)17-s + ⋯
L(s)  = 1  + (0.522 + 0.852i)2-s + (0.999 + 0.0318i)3-s + (−0.453 + 0.891i)4-s + (−0.830 + 0.556i)5-s + (0.495 + 0.868i)6-s + (−0.999 + 0.0159i)7-s + (−0.996 + 0.0796i)8-s + (0.997 + 0.0637i)9-s + (−0.908 − 0.417i)10-s + (0.0875 + 0.996i)11-s + (−0.481 + 0.876i)12-s + (0.839 + 0.543i)13-s + (−0.536 − 0.843i)14-s + (−0.848 + 0.529i)15-s + (−0.589 − 0.808i)16-s + (0.687 + 0.726i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.936 - 0.351i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.936 - 0.351i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $-0.936 - 0.351i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (381, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ -0.936 - 0.351i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4408949701 + 2.425082427i\)
\(L(\frac12)\) \(\approx\) \(-0.4408949701 + 2.425082427i\)
\(L(1)\) \(\approx\) \(0.9652892103 + 1.181547420i\)
\(L(1)\) \(\approx\) \(0.9652892103 + 1.181547420i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (0.522 + 0.852i)T \)
3 \( 1 + (0.999 + 0.0318i)T \)
5 \( 1 + (-0.830 + 0.556i)T \)
7 \( 1 + (-0.999 + 0.0159i)T \)
11 \( 1 + (0.0875 + 0.996i)T \)
13 \( 1 + (0.839 + 0.543i)T \)
17 \( 1 + (0.687 + 0.726i)T \)
19 \( 1 + (0.721 + 0.692i)T \)
23 \( 1 + (0.812 - 0.582i)T \)
29 \( 1 + (-0.522 + 0.852i)T \)
31 \( 1 + (-0.991 - 0.127i)T \)
37 \( 1 + (-0.639 + 0.768i)T \)
41 \( 1 + (-0.601 + 0.798i)T \)
43 \( 1 + (0.336 + 0.941i)T \)
47 \( 1 + (0.305 + 0.952i)T \)
53 \( 1 + (0.150 - 0.988i)T \)
59 \( 1 + (0.959 - 0.283i)T \)
61 \( 1 + (0.880 + 0.474i)T \)
67 \( 1 + (0.981 + 0.190i)T \)
71 \( 1 + (0.260 - 0.965i)T \)
73 \( 1 + (0.576 + 0.817i)T \)
79 \( 1 - T \)
83 \( 1 + (0.975 + 0.221i)T \)
89 \( 1 + (-0.939 - 0.343i)T \)
97 \( 1 + (-0.999 + 0.0159i)T \)
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

−18.251175648797862337101569001080, −17.02426630703600852552022284914, −16.06717158661112544068108870640, −15.68442391440791979559535020798, −15.17871439388137419028464166572, −14.15502896630671072121931253426, −13.585585668288547355865035000460, −13.15779923570281355289733195640, −12.53077825014422319732083177406, −11.7929075803218954803033928760, −11.10272351084298612229124296898, −10.36127802105368168125278598762, −9.46962578570154821267837893339, −9.01088469445395738884518676342, −8.49130180216158029263329868790, −7.46070620270253165374656729374, −6.858903634797544484351144718540, −5.594315001088231700569644466069, −5.25087375252606230020496931981, −3.92957573957183703965034616425, −3.58957180979030817713785553261, −3.161762392460566916265565229909, −2.28609944369491727015556879027, −1.05739747039964905780706910118, −0.56144242676770542915859678203, 1.32244218843785736854954177782, 2.51219977308002903257700459027, 3.41020557967439981917324399856, 3.613107870867398695125518282083, 4.34335366630365999292293211912, 5.28719505728928482177345449273, 6.37922262103217487501507101917, 6.885167194922764478097987950256, 7.4134465323795998883729649435, 8.15245969955736700831615488638, 8.757373871799761473844122917849, 9.56058958229956770064798976026, 10.135300712292431080782871898446, 11.17897652575086322568567999090, 12.11936392587395281899723305648, 12.771114871402909126234538735979, 13.09419043941892768805443706575, 14.16915532525403335956739560804, 14.549329929296632125528823346678, 15.106136471108095765443878158118, 15.76182477076048442150057846569, 16.29256419486261963410680579204, 16.819185471243254043041290693650, 18.04899000763892989397828911668, 18.58512742177453662931545688871

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