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Properties

Label 1-571-571.161-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

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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: $\chi_{571} (161, \cdot )$
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 \( 1 + (-0.627 + 0.778i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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