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Properties

Label 1-571-571.225-r0-0-0
Degree $1$
Conductor $571$
Sign $-0.622 - 0.782i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.789 − 0.614i)2-s + (−0.995 + 0.0990i)3-s + (0.245 − 0.969i)4-s + (−0.518 + 0.854i)5-s + (−0.724 + 0.689i)6-s + (0.601 − 0.799i)7-s + (−0.401 − 0.915i)8-s + (0.980 − 0.197i)9-s + (0.115 + 0.993i)10-s + (0.115 − 0.993i)11-s + (−0.148 + 0.988i)12-s + (−0.934 + 0.355i)13-s + (−0.0165 − 0.999i)14-s + (0.431 − 0.901i)15-s + (−0.879 − 0.475i)16-s + (0.991 − 0.131i)17-s + ⋯
L(s)  = 1  + (0.789 − 0.614i)2-s + (−0.995 + 0.0990i)3-s + (0.245 − 0.969i)4-s + (−0.518 + 0.854i)5-s + (−0.724 + 0.689i)6-s + (0.601 − 0.799i)7-s + (−0.401 − 0.915i)8-s + (0.980 − 0.197i)9-s + (0.115 + 0.993i)10-s + (0.115 − 0.993i)11-s + (−0.148 + 0.988i)12-s + (−0.934 + 0.355i)13-s + (−0.0165 − 0.999i)14-s + (0.431 − 0.901i)15-s + (−0.879 − 0.475i)16-s + (0.991 − 0.131i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.622 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.622 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.622 - 0.782i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (225, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ -0.622 - 0.782i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5336848884 - 1.107008444i\)
\(L(\frac12)\) \(\approx\) \(0.5336848884 - 1.107008444i\)
\(L(1)\) \(\approx\) \(0.9473154078 - 0.5462671608i\)
\(L(1)\) \(\approx\) \(0.9473154078 - 0.5462671608i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (0.789 - 0.614i)T \)
3 \( 1 + (-0.995 + 0.0990i)T \)
5 \( 1 + (-0.518 + 0.854i)T \)
7 \( 1 + (0.601 - 0.799i)T \)
11 \( 1 + (0.115 - 0.993i)T \)
13 \( 1 + (-0.934 + 0.355i)T \)
17 \( 1 + (0.991 - 0.131i)T \)
19 \( 1 + (-0.213 + 0.976i)T \)
23 \( 1 + (0.746 + 0.665i)T \)
29 \( 1 + (0.546 - 0.837i)T \)
31 \( 1 + (-0.879 - 0.475i)T \)
37 \( 1 + (0.0495 - 0.998i)T \)
41 \( 1 + (-0.0825 - 0.996i)T \)
43 \( 1 + (0.490 - 0.871i)T \)
47 \( 1 + (-0.879 - 0.475i)T \)
53 \( 1 + (-0.999 - 0.0330i)T \)
59 \( 1 + (0.945 + 0.324i)T \)
61 \( 1 + (-0.340 - 0.940i)T \)
67 \( 1 + (-0.999 - 0.0330i)T \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + (0.0495 - 0.998i)T \)
79 \( 1 + (0.997 + 0.0660i)T \)
83 \( 1 + (-0.724 + 0.689i)T \)
89 \( 1 + (-0.724 + 0.689i)T \)
97 \( 1 + (-0.768 + 0.639i)T \)
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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

−23.61585468396740582770667513389, −22.84567277784477618595663513926, −22.03356745632694780875772011747, −21.32206970605418509025857390925, −20.51387729542468499091919663444, −19.45755483191558698139888383961, −18.14463348819322158331684291137, −17.46449960570998891007422886919, −16.76526145465344271012289263118, −15.98266179624512102084466589835, −15.11651485101645900087114182996, −14.56261566191796608047197679940, −12.87100022922799064758216683924, −12.61053516239903192232797887264, −11.877134769982720111255221674187, −11.122560879358760087622477847, −9.68412646559308612158696009969, −8.53287712488359588450310288561, −7.60439129539308862845790965650, −6.84111711490254746555882848543, −5.63349231301157780804573045306, −4.78699944824036523612771694785, −4.62496446447726527626998614416, −2.91473885892876765776190567959, −1.475651786860813399884737362038, 0.57786073932783259577952864296, 1.90237299064807336688927322034, 3.42010366971677929294767664659, 4.04570055061653024531086292086, 5.13292534184818204771788229846, 5.9723334285008810199186940447, 6.99633366013528055409916591545, 7.7397574786486063370107239973, 9.605916337390473281207659208084, 10.48078901695754874818065557938, 11.03534071262805086741090235472, 11.75033325663677653282357063160, 12.42429505370941533712431650369, 13.69718720047482792709567842922, 14.39080371260758985943575499309, 15.14584480077948518767435835723, 16.243527267549461488120252971521, 16.9761697836872987857088129872, 18.08859648604689356232917262549, 19.028015905896314535481820442736, 19.42681428761683078778464853569, 20.84440900612024319144888091808, 21.35855229344262533580039989377, 22.24646077025518304756697056264, 22.8800229963069803089705388678

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