↑

Properties

Label 1-571-571.266-r1-0-0
Degree $1$
Conductor $571$
Sign $-0.960 - 0.276i$
Analytic cond. $61.3624$
Root an. cond. $61.3624$
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  + (0.879 − 0.475i)2-s + (0.518 + 0.854i)3-s + (0.546 − 0.837i)4-s + (0.601 − 0.799i)5-s + (0.863 + 0.504i)6-s + (−0.371 − 0.928i)7-s + (0.0825 − 0.996i)8-s + (−0.461 + 0.887i)9-s + (0.148 − 0.988i)10-s + (−0.148 − 0.988i)11-s + (0.999 + 0.0330i)12-s + (−0.909 + 0.416i)13-s + (−0.768 − 0.639i)14-s + (0.995 + 0.0990i)15-s + (−0.401 − 0.915i)16-s + (−0.746 + 0.665i)17-s + ⋯
L(s)  = 1  + (0.879 − 0.475i)2-s + (0.518 + 0.854i)3-s + (0.546 − 0.837i)4-s + (0.601 − 0.799i)5-s + (0.863 + 0.504i)6-s + (−0.371 − 0.928i)7-s + (0.0825 − 0.996i)8-s + (−0.461 + 0.887i)9-s + (0.148 − 0.988i)10-s + (−0.148 − 0.988i)11-s + (0.999 + 0.0330i)12-s + (−0.909 + 0.416i)13-s + (−0.768 − 0.639i)14-s + (0.995 + 0.0990i)15-s + (−0.401 − 0.915i)16-s + (−0.746 + 0.665i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.960 - 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.960 - 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.960 - 0.276i$
Analytic conductor: \(61.3624\)
Root analytic conductor: \(61.3624\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (266, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (1:\ ),\ -0.960 - 0.276i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3435574271 - 2.433435925i\)
\(L(\frac12)\) \(\approx\) \(0.3435574271 - 2.433435925i\)
\(L(1)\) \(\approx\) \(1.561250553 - 0.7765190629i\)
\(L(1)\) \(\approx\) \(1.561250553 - 0.7765190629i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (0.879 - 0.475i)T \)
3 \( 1 + (0.518 + 0.854i)T \)
5 \( 1 + (0.601 - 0.799i)T \)
7 \( 1 + (-0.371 - 0.928i)T \)
11 \( 1 + (-0.148 - 0.988i)T \)
13 \( 1 + (-0.909 + 0.416i)T \)
17 \( 1 + (-0.746 + 0.665i)T \)
19 \( 1 + (-0.922 - 0.386i)T \)
23 \( 1 + (0.652 + 0.757i)T \)
29 \( 1 + (-0.677 - 0.735i)T \)
31 \( 1 + (-0.401 - 0.915i)T \)
37 \( 1 + (0.490 + 0.871i)T \)
41 \( 1 + (-0.945 - 0.324i)T \)
43 \( 1 + (0.894 + 0.446i)T \)
47 \( 1 + (0.401 + 0.915i)T \)
53 \( 1 + (-0.180 + 0.983i)T \)
59 \( 1 + (0.245 - 0.969i)T \)
61 \( 1 + (0.431 - 0.901i)T \)
67 \( 1 + (-0.180 + 0.983i)T \)
71 \( 1 + (0.809 - 0.587i)T \)
73 \( 1 + (-0.490 - 0.871i)T \)
79 \( 1 + (0.934 + 0.355i)T \)
83 \( 1 + (0.863 + 0.504i)T \)
89 \( 1 + (-0.863 - 0.504i)T \)
97 \( 1 + (-0.627 - 0.778i)T \)
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

−23.322539943884231616773456768549, −22.62602129036254136795282329415, −22.01173963008049085984186717067, −21.05611894159603969823553534352, −20.18396166362392301598912534584, −19.27390117909760205910018371015, −18.20557821979547306503817990188, −17.75885187722194752741305725693, −16.70716980562657650457431996358, −15.324406481340338705116974614176, −14.867651051492689088662140908429, −14.272289671898154620364085637567, −13.10692450711674139387504363349, −12.67339642389512786922255506018, −11.88088719656913455368769701081, −10.66261056887314316494637779164, −9.39020851197611045727075848939, −8.50436623772519792470346328783, −7.186765668771307152852410951720, −6.90821667872505369170062857144, −5.88585794569612670785389722839, −4.97254107897809263818834623999, −3.47324708880876806025916756514, −2.39128638166044096359388653368, −2.18538582638746994116030494168, 0.34162267635421790447344740657, 1.823931527703697668548561776731, 2.83521678100476711619400543694, 3.98735030026208885775732688308, 4.54545459052538219027506202532, 5.54559095801626670759619787952, 6.50555371260259113627715610936, 7.88071659889433558532623751402, 9.15256255825975700001504864717, 9.723691040056812398981382822319, 10.69892726772486543764439445300, 11.32799089077336502142941068494, 12.77915416462164170484996359892, 13.39611619540215406216547063780, 13.958996650510716542506723014, 14.95214667172957871244976894229, 15.75264722688956606246288180793, 16.79369684232564765091712129701, 17.12948018604376600765991627589, 19.1159111369301095633583012159, 19.5288707557382542830312505003, 20.46112588721224444157853020623, 20.969612793053062752945032547643, 21.89975398219437355215454117427, 22.17663703922537225070179405962

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