↑

Properties

Label 1-571-571.83-r0-0-0
Degree $1$
Conductor $571$
Sign $0.480 + 0.877i$
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  + (−0.191 − 0.981i)2-s + (−0.256 + 0.966i)3-s + (−0.926 + 0.376i)4-s + (0.802 − 0.596i)5-s + (0.997 + 0.0660i)6-s + (−0.148 + 0.988i)7-s + (0.546 + 0.837i)8-s + (−0.868 − 0.495i)9-s + (−0.739 − 0.673i)10-s + (0.952 + 0.303i)11-s + (−0.126 − 0.991i)12-s + (−0.997 − 0.0770i)13-s + (0.999 − 0.0440i)14-s + (0.371 + 0.928i)15-s + (0.716 − 0.697i)16-s + (−0.170 + 0.985i)17-s + ⋯
L(s)  = 1  + (−0.191 − 0.981i)2-s + (−0.256 + 0.966i)3-s + (−0.926 + 0.376i)4-s + (0.802 − 0.596i)5-s + (0.997 + 0.0660i)6-s + (−0.148 + 0.988i)7-s + (0.546 + 0.837i)8-s + (−0.868 − 0.495i)9-s + (−0.739 − 0.673i)10-s + (0.952 + 0.303i)11-s + (−0.126 − 0.991i)12-s + (−0.997 − 0.0770i)13-s + (0.999 − 0.0440i)14-s + (0.371 + 0.928i)15-s + (0.716 − 0.697i)16-s + (−0.170 + 0.985i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.480 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.480 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.480 + 0.877i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ 0.480 + 0.877i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8196910628 + 0.4855711304i\)
\(L(\frac12)\) \(\approx\) \(0.8196910628 + 0.4855711304i\)
\(L(1)\) \(\approx\) \(0.8623548949 + 0.03635987066i\)
\(L(1)\) \(\approx\) \(0.8623548949 + 0.03635987066i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (-0.191 - 0.981i)T \)
3 \( 1 + (-0.256 + 0.966i)T \)
5 \( 1 + (0.802 - 0.596i)T \)
7 \( 1 + (-0.148 + 0.988i)T \)
11 \( 1 + (0.952 + 0.303i)T \)
13 \( 1 + (-0.997 - 0.0770i)T \)
17 \( 1 + (-0.170 + 0.985i)T \)
19 \( 1 + (0.840 + 0.542i)T \)
23 \( 1 + (-0.627 - 0.778i)T \)
29 \( 1 + (0.0275 + 0.999i)T \)
31 \( 1 + (0.245 + 0.969i)T \)
37 \( 1 + (-0.381 - 0.924i)T \)
41 \( 1 + (-0.298 + 0.954i)T \)
43 \( 1 + (0.202 + 0.979i)T \)
47 \( 1 + (-0.962 - 0.272i)T \)
53 \( 1 + (-0.421 - 0.906i)T \)
59 \( 1 + (-0.986 + 0.164i)T \)
61 \( 1 + (0.391 + 0.920i)T \)
67 \( 1 + (0.996 + 0.0880i)T \)
71 \( 1 + (0.669 + 0.743i)T \)
73 \( 1 + (-0.609 + 0.792i)T \)
79 \( 1 + (0.984 + 0.175i)T \)
83 \( 1 + (-0.556 + 0.831i)T \)
89 \( 1 + (-0.441 - 0.897i)T \)
97 \( 1 + (0.970 + 0.240i)T \)
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

−23.18835276158765687495884267178, −22.36131847361550597803723224070, −22.08424280461428961136644345707, −20.3723287652310252614215781455, −19.44790932409317661448235820328, −18.78284295560198446415913001606, −17.83001095476321587595881384844, −17.21468755685060345833897282246, −16.82888222295357469919668258788, −15.552666982174605919084118945002, −14.35483843081542934029843438889, −13.77152282804796266488912439208, −13.47259366372395689801604215834, −12.073168825449716696748293981, −11.09050767098327968173369349576, −9.85551992112680929697026311611, −9.32696354667926931932896405449, −7.87615761834643459435781659006, −7.16687627465261058693065656152, −6.618734685562578087858722156115, −5.746205208622066424881760446827, −4.7208176329446863447796813379, −3.271383960434670187329921739467, −1.823530922340473559667589295144, −0.57091576127743195940894819590, 1.43610114777747489200412559326, 2.49398478371706712551332201215, 3.58993397206183422132110298927, 4.70127712815255239911785247803, 5.343993746277408414906216860876, 6.37748343040666458924664834623, 8.3043567407515790033428286432, 8.99322538775296721543540640365, 9.73345117445287633500737055040, 10.22419127070787904962415331374, 11.44339804338784110989253240857, 12.29029011658489521496360519598, 12.72223079374516562182433139029, 14.30537426627197855100665629167, 14.60828830424537052435992386118, 16.073253966859076102559375123224, 16.81683976055239474707373780681, 17.61174294472002681141207985724, 18.22285712103007065401023663403, 19.64342444927828731952136737461, 20.02024532912805307964889526902, 21.10800502400395730529528607937, 21.66678831000659240720386886806, 22.18917786442119585492871532304, 22.86124362422882973840188984570

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