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Properties

Label 1-571-571.45-r0-0-0
Degree $1$
Conductor $571$
Sign $0.0814 - 0.996i$
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  + (−0.821 + 0.569i)2-s + (−0.0385 − 0.999i)3-s + (0.350 − 0.936i)4-s + (0.840 − 0.542i)5-s + (0.601 + 0.799i)6-s + (0.490 − 0.871i)7-s + (0.245 + 0.969i)8-s + (−0.997 + 0.0770i)9-s + (−0.381 + 0.924i)10-s + (−0.609 + 0.792i)11-s + (−0.949 − 0.314i)12-s + (0.528 − 0.849i)13-s + (0.0935 + 0.995i)14-s + (−0.574 − 0.818i)15-s + (−0.754 − 0.656i)16-s + (0.224 + 0.974i)17-s + ⋯
L(s)  = 1  + (−0.821 + 0.569i)2-s + (−0.0385 − 0.999i)3-s + (0.350 − 0.936i)4-s + (0.840 − 0.542i)5-s + (0.601 + 0.799i)6-s + (0.490 − 0.871i)7-s + (0.245 + 0.969i)8-s + (−0.997 + 0.0770i)9-s + (−0.381 + 0.924i)10-s + (−0.609 + 0.792i)11-s + (−0.949 − 0.314i)12-s + (0.528 − 0.849i)13-s + (0.0935 + 0.995i)14-s + (−0.574 − 0.818i)15-s + (−0.754 − 0.656i)16-s + (0.224 + 0.974i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0814 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0814 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.0814 - 0.996i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ 0.0814 - 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8199130422 - 0.7556278344i\)
\(L(\frac12)\) \(\approx\) \(0.8199130422 - 0.7556278344i\)
\(L(1)\) \(\approx\) \(0.8402347900 - 0.3049839582i\)
\(L(1)\) \(\approx\) \(0.8402347900 - 0.3049839582i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (-0.821 + 0.569i)T \)
3 \( 1 + (-0.0385 - 0.999i)T \)
5 \( 1 + (0.840 - 0.542i)T \)
7 \( 1 + (0.490 - 0.871i)T \)
11 \( 1 + (-0.609 + 0.792i)T \)
13 \( 1 + (0.528 - 0.849i)T \)
17 \( 1 + (0.224 + 0.974i)T \)
19 \( 1 + (0.938 - 0.345i)T \)
23 \( 1 + (0.997 - 0.0660i)T \)
29 \( 1 + (0.137 - 0.990i)T \)
31 \( 1 + (0.945 - 0.324i)T \)
37 \( 1 + (0.970 - 0.240i)T \)
41 \( 1 + (-0.998 - 0.0550i)T \)
43 \( 1 + (-0.234 - 0.972i)T \)
47 \( 1 + (-0.191 + 0.981i)T \)
53 \( 1 + (0.329 + 0.944i)T \)
59 \( 1 + (-0.677 - 0.735i)T \)
61 \( 1 + (-0.126 + 0.991i)T \)
67 \( 1 + (-0.982 - 0.186i)T \)
71 \( 1 + (-0.978 - 0.207i)T \)
73 \( 1 + (-0.693 - 0.720i)T \)
79 \( 1 + (0.930 + 0.366i)T \)
83 \( 1 + (-0.992 + 0.120i)T \)
89 \( 1 + (0.391 - 0.920i)T \)
97 \( 1 + (0.266 + 0.963i)T \)
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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

−23.16695968696339048932119244617, −22.14342945874092589417637236981, −21.52381396422293288710448782969, −21.09012402976866623973876037814, −20.390576877210201693071140158, −19.07120124669459498832717078381, −18.36748738778106520549402205089, −17.85198842698184586197396601407, −16.652199936149794084303091519837, −16.1658394888359681417793185146, −15.16700219424897876871555094560, −14.14193543643501333068498968360, −13.32657772903470925375544758426, −11.80044030286091930659391404259, −11.328750759470126364304978327051, −10.48991247692673290056957026957, −9.62084620244267103607638665719, −8.98379632702020103585424237472, −8.19172657412264082398133929376, −6.81855576108663123964515301809, −5.69735154208719438110232090939, −4.80006040733078795103759758321, −3.18245721546162164036541025394, −2.79754906801703425101807287855, −1.42719398843071065004884387412, 0.859359515840556728975468015919, 1.555156826586416134608454916072, 2.70039205379885467667941105426, 4.733040308302346019029460230446, 5.62950141610630242182316957643, 6.441590632430715654332433383392, 7.52647678129259832001301536327, 8.01408185717123686392518327715, 8.99119312480559702743902729695, 10.11176947285423774528864965560, 10.74556199363539809607456781721, 11.922431222317176314628426846999, 13.17331111671730521127176202448, 13.560529960853610240630250630059, 14.618398192886551369274690774503, 15.52082868915744096803386922110, 16.799012694535158263759358299629, 17.33706564438020762203913963400, 17.84767781863418670732648031828, 18.5867089893946850405933293335, 19.67458085186646853122835426213, 20.415859482802629849169409281308, 20.931157921759316223986961759993, 22.618172137849848226725387601803, 23.419684644163689698356294003990

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