↑

Properties

Label 1-571-571.226-r0-0-0
Degree $1$
Conductor $571$
Sign $0.281 - 0.959i$
Analytic cond. $2.65171$
Root an. cond. $2.65171$
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  + (0.546 + 0.837i)2-s + (−0.461 − 0.887i)3-s + (−0.401 + 0.915i)4-s + (−0.277 + 0.960i)5-s + (0.490 − 0.871i)6-s + (−0.724 − 0.689i)7-s + (−0.986 + 0.164i)8-s + (−0.574 + 0.818i)9-s + (−0.956 + 0.293i)10-s + (−0.956 − 0.293i)11-s + (0.997 − 0.0660i)12-s + (0.652 + 0.757i)13-s + (0.180 − 0.983i)14-s + (0.980 − 0.197i)15-s + (−0.677 − 0.735i)16-s + (0.115 + 0.993i)17-s + ⋯
L(s)  = 1  + (0.546 + 0.837i)2-s + (−0.461 − 0.887i)3-s + (−0.401 + 0.915i)4-s + (−0.277 + 0.960i)5-s + (0.490 − 0.871i)6-s + (−0.724 − 0.689i)7-s + (−0.986 + 0.164i)8-s + (−0.574 + 0.818i)9-s + (−0.956 + 0.293i)10-s + (−0.956 − 0.293i)11-s + (0.997 − 0.0660i)12-s + (0.652 + 0.757i)13-s + (0.180 − 0.983i)14-s + (0.980 − 0.197i)15-s + (−0.677 − 0.735i)16-s + (0.115 + 0.993i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.281 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.281 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.281 - 0.959i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ 0.281 - 0.959i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3716509245 - 0.2783432428i\)
\(L(\frac12)\) \(\approx\) \(0.3716509245 - 0.2783432428i\)
\(L(1)\) \(\approx\) \(0.7548477691 + 0.1800565548i\)
\(L(1)\) \(\approx\) \(0.7548477691 + 0.1800565548i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (0.546 + 0.837i)T \)
3 \( 1 + (-0.461 - 0.887i)T \)
5 \( 1 + (-0.277 + 0.960i)T \)
7 \( 1 + (-0.724 - 0.689i)T \)
11 \( 1 + (-0.956 - 0.293i)T \)
13 \( 1 + (0.652 + 0.757i)T \)
17 \( 1 + (0.115 + 0.993i)T \)
19 \( 1 + (0.701 - 0.712i)T \)
23 \( 1 + (-0.148 - 0.988i)T \)
29 \( 1 + (-0.0825 - 0.996i)T \)
31 \( 1 + (-0.677 - 0.735i)T \)
37 \( 1 + (-0.518 - 0.854i)T \)
41 \( 1 + (0.789 - 0.614i)T \)
43 \( 1 + (0.601 - 0.799i)T \)
47 \( 1 + (-0.677 - 0.735i)T \)
53 \( 1 + (-0.934 + 0.355i)T \)
59 \( 1 + (-0.879 + 0.475i)T \)
61 \( 1 + (-0.627 + 0.778i)T \)
67 \( 1 + (-0.934 + 0.355i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (-0.518 - 0.854i)T \)
79 \( 1 + (0.746 - 0.665i)T \)
83 \( 1 + (0.490 - 0.871i)T \)
89 \( 1 + (0.490 - 0.871i)T \)
97 \( 1 + (-0.213 - 0.976i)T \)
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

−23.09741073852179070019449632207, −22.61402405677376220047021624647, −21.66560343068328749607376508308, −20.890021630138912123584099607403, −20.3962552532616418248629901277, −19.61123603050380313278678547498, −18.371161309937123419722086572060, −17.81049425095278274094471723262, −16.25665801633127758171334517584, −15.88585925852861900818395773853, −15.18526502561623449112863654241, −13.95208433703408953582885767854, −12.874951365943360951833017766415, −12.375723670507435812036548533023, −11.50548835160871346942831364195, −10.62341209934116872340934038444, −9.599163884251133923358766596295, −9.21861570767266793582456516068, −7.96287470601783476986978256495, −6.173895538136589831512157946674, −5.28177815505187794886823464056, −4.94228954105072597130394293991, −3.54466994497224783736138637776, −2.99922269469202608190409186188, −1.28299096044300839391252245997, 0.22181710922833196498738229293, 2.32580974141677076103106966083, 3.36591944274063768630508541361, 4.355726032833663327059554388980, 5.83975944258382392458803250719, 6.299695877395210933455965793465, 7.27256588251335807349636151093, 7.707788789336119579543655437714, 8.890635222396497418787563532704, 10.41956942901941297540927751974, 11.1699589863655526333351822535, 12.19047034919599718698868456943, 13.14616594771798049285139613352, 13.66964205011554928104515212635, 14.45480821617118966974626268779, 15.624147403549657127371247710206, 16.28212398843158574146508662421, 17.142309314073533761288860073271, 18.03890745568095789038131298133, 18.734086021442950909062888647984, 19.43865899101072120460944219581, 20.74114846191536431706319007729, 21.84683665114613308775743295026, 22.57816433874922747134610401967, 23.1796841833129646702602981373

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