↑

Properties

Label 1-29e2-841.50-r1-0-0
Degree $1$
Conductor $841$
Sign $0.0320 + 0.999i$
Analytic cond. $90.3780$
Root an. cond. $90.3780$
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  + (−0.0309 − 0.999i)2-s + (0.791 + 0.611i)3-s + (−0.998 + 0.0618i)4-s + (0.997 + 0.0773i)5-s + (0.586 − 0.809i)6-s + (0.598 + 0.800i)7-s + (0.0927 + 0.995i)8-s + (0.252 + 0.967i)9-s + (0.0464 − 0.998i)10-s + (−0.592 + 0.805i)11-s + (−0.827 − 0.561i)12-s + (0.0695 + 0.997i)13-s + (0.781 − 0.623i)14-s + (0.741 + 0.670i)15-s + (0.992 − 0.123i)16-s + (−0.928 + 0.370i)17-s + ⋯
L(s)  = 1  + (−0.0309 − 0.999i)2-s + (0.791 + 0.611i)3-s + (−0.998 + 0.0618i)4-s + (0.997 + 0.0773i)5-s + (0.586 − 0.809i)6-s + (0.598 + 0.800i)7-s + (0.0927 + 0.995i)8-s + (0.252 + 0.967i)9-s + (0.0464 − 0.998i)10-s + (−0.592 + 0.805i)11-s + (−0.827 − 0.561i)12-s + (0.0695 + 0.997i)13-s + (0.781 − 0.623i)14-s + (0.741 + 0.670i)15-s + (0.992 − 0.123i)16-s + (−0.928 + 0.370i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0320 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0320 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(841\)    =    \(29^{2}\)
Sign: $0.0320 + 0.999i$
Analytic conductor: \(90.3780\)
Root analytic conductor: \(90.3780\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{841} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 841,\ (1:\ ),\ 0.0320 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.886640944 + 1.827179548i\)
\(L(\frac12)\) \(\approx\) \(1.886640944 + 1.827179548i\)
\(L(1)\) \(\approx\) \(1.438802141 + 0.1340863466i\)
\(L(1)\) \(\approx\) \(1.438802141 + 0.1340863466i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 \)
good2 \( 1 + (-0.0309 - 0.999i)T \)
3 \( 1 + (0.791 + 0.611i)T \)
5 \( 1 + (0.997 + 0.0773i)T \)
7 \( 1 + (0.598 + 0.800i)T \)
11 \( 1 + (-0.592 + 0.805i)T \)
13 \( 1 + (0.0695 + 0.997i)T \)
17 \( 1 + (-0.928 + 0.370i)T \)
19 \( 1 + (-0.475 - 0.879i)T \)
23 \( 1 + (0.969 + 0.245i)T \)
31 \( 1 + (-0.447 - 0.894i)T \)
37 \( 1 + (-0.199 - 0.979i)T \)
41 \( 1 + iT \)
43 \( 1 + (0.488 + 0.872i)T \)
47 \( 1 + (0.934 + 0.355i)T \)
53 \( 1 + (0.495 + 0.868i)T \)
59 \( 1 + (-0.161 - 0.986i)T \)
61 \( 1 + (-0.567 + 0.823i)T \)
67 \( 1 + (-0.715 - 0.698i)T \)
71 \( 1 + (0.0386 - 0.999i)T \)
73 \( 1 + (-0.917 - 0.398i)T \)
79 \( 1 + (-0.230 + 0.973i)T \)
83 \( 1 + (0.832 + 0.554i)T \)
89 \( 1 + (-0.971 - 0.237i)T \)
97 \( 1 + (-0.994 - 0.100i)T \)
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

−21.7247823421409592357592231671, −20.85186941676544894888780264950, −20.27814404003437367881621838318, −19.05058371561196457463781126422, −18.3881301858209733726435079009, −17.644451464372864249793060961304, −17.1354258717048353934424710058, −16.10183200437893279763923195499, −15.097785154482741428789281427468, −14.41651730736703437032447495388, −13.580057905654963213823968231765, −13.34466026710814594506467151269, −12.42974958186888102238828354535, −10.71508341256385735965702245120, −10.13354676450572703953296163239, −8.83663141817561777727669835214, −8.51311243369299506518376962764, −7.49202819692230840513030691382, −6.8120283063619452010057832591, −5.82857177209616060678585902734, −5.02435238750221786301356898433, −3.8149830493304953508043676366, −2.73469275679489117435148137603, −1.42462724218846049640865676, −0.49021096250641641492365404036, 1.6134831168927313702708856668, 2.25983351901204470368733697227, 2.828443045096488037189886284574, 4.35711226796130747489232856416, 4.77829432199782890831966077148, 5.836238867847294605927814967486, 7.314889839685940340793110672991, 8.51394347130172231150006402883, 9.21456688651810503416541168687, 9.54788077968192536374570392277, 10.7854985982204710806863189001, 11.12915324067653406004136326009, 12.51027048881428752653734304820, 13.22057748917939671727360226495, 13.90293607319407923401544456574, 14.82651630552323421013813130812, 15.29839942482205375618141915142, 16.69411907782687365358895179136, 17.60256278732831432791790918891, 18.23835214077611179623731883277, 19.06631112255810317477945269994, 19.85076289482338626686443773831, 20.825998947024249006212533299266, 21.19188757622410212630550518063, 21.79401264058943548164623890058

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