↑

Properties

Label 1-1441-1441.268-r1-0-0
Degree $1$
Conductor $1441$
Sign $-0.358 + 0.933i$
Analytic cond. $154.856$
Root an. cond. $154.856$
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  + (0.0724 − 0.997i)2-s + (0.981 + 0.192i)3-s + (−0.989 − 0.144i)4-s + (−0.681 − 0.732i)5-s + (0.262 − 0.964i)6-s + (−0.354 + 0.935i)7-s + (−0.215 + 0.976i)8-s + (0.926 + 0.377i)9-s + (−0.779 + 0.626i)10-s + (−0.943 − 0.331i)12-s + (−0.354 + 0.935i)13-s + (0.906 + 0.421i)14-s + (−0.527 − 0.849i)15-s + (0.958 + 0.285i)16-s + (0.943 − 0.331i)17-s + (0.443 − 0.896i)18-s + ⋯
L(s)  = 1  + (0.0724 − 0.997i)2-s + (0.981 + 0.192i)3-s + (−0.989 − 0.144i)4-s + (−0.681 − 0.732i)5-s + (0.262 − 0.964i)6-s + (−0.354 + 0.935i)7-s + (−0.215 + 0.976i)8-s + (0.926 + 0.377i)9-s + (−0.779 + 0.626i)10-s + (−0.943 − 0.331i)12-s + (−0.354 + 0.935i)13-s + (0.906 + 0.421i)14-s + (−0.527 − 0.849i)15-s + (0.958 + 0.285i)16-s + (0.943 − 0.331i)17-s + (0.443 − 0.896i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.358 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.358 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.358 + 0.933i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (268, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ -0.358 + 0.933i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3562990669 + 0.5183347964i\)
\(L(\frac12)\) \(\approx\) \(0.3562990669 + 0.5183347964i\)
\(L(1)\) \(\approx\) \(1.011217741 - 0.3025905633i\)
\(L(1)\) \(\approx\) \(1.011217741 - 0.3025905633i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.0724 - 0.997i)T \)
3 \( 1 + (0.981 + 0.192i)T \)
5 \( 1 + (-0.681 - 0.732i)T \)
7 \( 1 + (-0.354 + 0.935i)T \)
13 \( 1 + (-0.354 + 0.935i)T \)
17 \( 1 + (0.943 - 0.331i)T \)
19 \( 1 + (-0.0241 + 0.999i)T \)
23 \( 1 + (0.861 - 0.506i)T \)
29 \( 1 + (0.970 + 0.239i)T \)
31 \( 1 + (-0.995 + 0.0965i)T \)
37 \( 1 + (-0.885 + 0.464i)T \)
41 \( 1 + (-0.607 + 0.794i)T \)
43 \( 1 + (-0.681 - 0.732i)T \)
47 \( 1 + (-0.926 - 0.377i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (-0.943 + 0.331i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 + (0.681 - 0.732i)T \)
71 \( 1 + (-0.399 - 0.916i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (0.906 - 0.421i)T \)
83 \( 1 + (0.989 - 0.144i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (-0.836 + 0.548i)T \)
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

−19.992748346807081964366397910916, −19.50480124461789983072077651785, −18.87516267860748483707702517409, −18.0016632882571357895454388, −17.30348508185595626756782941937, −16.35867190666717782398389564397, −15.51027529407436122497618666338, −15.08225758279262609546909148406, −14.32714924612198987279255532264, −13.697611343642187079569874067730, −12.9441773109907926086094287511, −12.23353131396119659988615844410, −10.79829383587301779070437224726, −10.10751174774584690895411947558, −9.308805601881360432923201499760, −8.30228483685811952995322741070, −7.65574442444132274211752566188, −7.13996853363569960316273899325, −6.507280731216254740949529715644, −5.20990938243133389176332473420, −4.21484673194903696581150632293, −3.40335789222342421418183413945, −2.92431865449326638010289504776, −1.13937513899710175368958106309, −0.10972041032389164439890807719, 1.320439824192056173366572442863, 2.061519085277469797308708052246, 3.192672958184094968858455711358, 3.614943310345736076358412373470, 4.76573676900034120857846651500, 5.21766687672723549315362625124, 6.72647426030252498667350804796, 7.97224834048213977785233822185, 8.502695717342076917256354796090, 9.24719106274863700474907642656, 9.745277040859310335498373318965, 10.72475530789041238288955413320, 11.95376464929230435213596843157, 12.19842638989600445148860219800, 12.98558090004239955382517977810, 13.85303713771551097308826547741, 14.655029947967138150725625266405, 15.22845135734430497059490965818, 16.33216101230298843085321851031, 16.78425279407408838422530430364, 18.33127934739630453984557748488, 18.76541199891056008323769164260, 19.3883225816529261848299508685, 19.969295768411618300589800889689, 20.83971305426726887515270448074

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