↑

Properties

Label 1-571-571.268-r1-0-0
Degree $1$
Conductor $571$
Sign $-0.611 + 0.791i$
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  + (0.0825 − 0.996i)2-s + (−0.371 − 0.928i)3-s + (−0.986 − 0.164i)4-s + (0.965 − 0.261i)5-s + (−0.956 + 0.293i)6-s + (−0.115 + 0.993i)7-s + (−0.245 + 0.969i)8-s + (−0.724 + 0.689i)9-s + (−0.180 − 0.983i)10-s + (0.180 − 0.983i)11-s + (0.213 + 0.976i)12-s + (−0.340 − 0.940i)13-s + (0.980 + 0.197i)14-s + (−0.601 − 0.799i)15-s + (0.945 + 0.324i)16-s + (0.0165 − 0.999i)17-s + ⋯
L(s)  = 1  + (0.0825 − 0.996i)2-s + (−0.371 − 0.928i)3-s + (−0.986 − 0.164i)4-s + (0.965 − 0.261i)5-s + (−0.956 + 0.293i)6-s + (−0.115 + 0.993i)7-s + (−0.245 + 0.969i)8-s + (−0.724 + 0.689i)9-s + (−0.180 − 0.983i)10-s + (0.180 − 0.983i)11-s + (0.213 + 0.976i)12-s + (−0.340 − 0.940i)13-s + (0.980 + 0.197i)14-s + (−0.601 − 0.799i)15-s + (0.945 + 0.324i)16-s + (0.0165 − 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.611 + 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.611 + 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.611 + 0.791i$
Analytic conductor: \(61.3624\)
Root analytic conductor: \(61.3624\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (268, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (1:\ ),\ -0.611 + 0.791i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.5836930005 - 1.188292618i\)
\(L(\frac12)\) \(\approx\) \(-0.5836930005 - 1.188292618i\)
\(L(1)\) \(\approx\) \(0.5663247698 - 0.7869070220i\)
\(L(1)\) \(\approx\) \(0.5663247698 - 0.7869070220i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (0.0825 - 0.996i)T \)
3 \( 1 + (-0.371 - 0.928i)T \)
5 \( 1 + (0.965 - 0.261i)T \)
7 \( 1 + (-0.115 + 0.993i)T \)
11 \( 1 + (0.180 - 0.983i)T \)
13 \( 1 + (-0.340 - 0.940i)T \)
17 \( 1 + (0.0165 - 0.999i)T \)
19 \( 1 + (0.846 + 0.533i)T \)
23 \( 1 + (-0.768 - 0.639i)T \)
29 \( 1 + (0.789 - 0.614i)T \)
31 \( 1 + (0.945 + 0.324i)T \)
37 \( 1 + (0.828 - 0.560i)T \)
41 \( 1 + (-0.546 - 0.837i)T \)
43 \( 1 + (0.991 + 0.131i)T \)
47 \( 1 + (-0.945 - 0.324i)T \)
53 \( 1 + (-0.922 + 0.386i)T \)
59 \( 1 + (-0.677 + 0.735i)T \)
61 \( 1 + (-0.518 - 0.854i)T \)
67 \( 1 + (-0.922 + 0.386i)T \)
71 \( 1 + (0.809 - 0.587i)T \)
73 \( 1 + (-0.828 + 0.560i)T \)
79 \( 1 + (-0.701 + 0.712i)T \)
83 \( 1 + (-0.956 + 0.293i)T \)
89 \( 1 + (0.956 - 0.293i)T \)
97 \( 1 + (-0.461 + 0.887i)T \)
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

−23.54201838612200326011641383706, −22.756695558290535395619337417782, −21.989079196354312229838214868231, −21.4636622185805026378420314772, −20.42228747729420760546213724155, −19.43127010202874584033521528682, −17.99154768567132983189241659853, −17.4887557126855720110797555322, −16.88616758570815973997871821094, −16.12565215224491330669009643684, −15.152146524295097818464822715505, −14.352309441028290026885946353, −13.79148835069949401521001410082, −12.74122318294806512470247756465, −11.53255765778732520142810717478, −10.1990493777030743260143640115, −9.83910376859174071974196147458, −9.06160441364587978566586852529, −7.71153840405198148485041006180, −6.66262946820154072160296961129, −6.11246685726793522138410524127, −4.84239350873284877779169014672, −4.357795949351292417998706465782, −3.17199558863322972487828681111, −1.34695421144235587166928886357, 0.361942176642462033172446498370, 1.28600878983732993946849811645, 2.48237991035032870549190217571, 2.97769163324351380773719225520, 4.87081261267683501610223038081, 5.67295484702451102823586792499, 6.21314104979687754664022174089, 7.90052747671191849267158160444, 8.68692372238494822989046207268, 9.62212487616679139412565872143, 10.51620324430209768034078915495, 11.626510965235405779733163685052, 12.221108751329993613887721585532, 12.94325951030768046734632548778, 13.864207022622103954505439583580, 14.282434061428342049106134377954, 15.8999329479747785351519578306, 16.96821106015885312210362355602, 17.92049389829794889073039857386, 18.27984219045146615140658721352, 19.0849760987519011360147567463, 19.978609441276554176537001906104, 20.860482034445493425320204636640, 21.76432251302860908632665555768, 22.39241706649092961966960916015

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