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Properties

Label 1-571-571.15-r1-0-0
Degree $1$
Conductor $571$
Sign $-0.449 - 0.893i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.945 + 0.324i)2-s + (−0.0495 − 0.998i)3-s + (0.789 − 0.614i)4-s + (0.490 + 0.871i)5-s + (0.371 + 0.928i)6-s + (−0.894 + 0.446i)7-s + (−0.546 + 0.837i)8-s + (−0.995 + 0.0990i)9-s + (−0.746 − 0.665i)10-s + (0.746 − 0.665i)11-s + (−0.652 − 0.757i)12-s + (0.180 + 0.983i)13-s + (0.701 − 0.712i)14-s + (0.846 − 0.533i)15-s + (0.245 − 0.969i)16-s + (−0.997 + 0.0660i)17-s + ⋯
L(s)  = 1  + (−0.945 + 0.324i)2-s + (−0.0495 − 0.998i)3-s + (0.789 − 0.614i)4-s + (0.490 + 0.871i)5-s + (0.371 + 0.928i)6-s + (−0.894 + 0.446i)7-s + (−0.546 + 0.837i)8-s + (−0.995 + 0.0990i)9-s + (−0.746 − 0.665i)10-s + (0.746 − 0.665i)11-s + (−0.652 − 0.757i)12-s + (0.180 + 0.983i)13-s + (0.701 − 0.712i)14-s + (0.846 − 0.533i)15-s + (0.245 − 0.969i)16-s + (−0.997 + 0.0660i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.449 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.449 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.449 - 0.893i$
Analytic conductor: \(61.3624\)
Root analytic conductor: \(61.3624\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (1:\ ),\ -0.449 - 0.893i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2060237350 - 0.3344647042i\)
\(L(\frac12)\) \(\approx\) \(0.2060237350 - 0.3344647042i\)
\(L(1)\) \(\approx\) \(0.5952858869 + 0.006779506969i\)
\(L(1)\) \(\approx\) \(0.5952858869 + 0.006779506969i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (-0.945 + 0.324i)T \)
3 \( 1 + (-0.0495 - 0.998i)T \)
5 \( 1 + (0.490 + 0.871i)T \)
7 \( 1 + (-0.894 + 0.446i)T \)
11 \( 1 + (0.746 - 0.665i)T \)
13 \( 1 + (0.180 + 0.983i)T \)
17 \( 1 + (-0.997 + 0.0660i)T \)
19 \( 1 + (0.627 + 0.778i)T \)
23 \( 1 + (-0.934 - 0.355i)T \)
29 \( 1 + (-0.879 + 0.475i)T \)
31 \( 1 + (0.245 - 0.969i)T \)
37 \( 1 + (-0.724 + 0.689i)T \)
41 \( 1 + (0.677 - 0.735i)T \)
43 \( 1 + (0.863 - 0.504i)T \)
47 \( 1 + (-0.245 + 0.969i)T \)
53 \( 1 + (0.0165 - 0.999i)T \)
59 \( 1 + (-0.986 - 0.164i)T \)
61 \( 1 + (-0.574 + 0.818i)T \)
67 \( 1 + (0.0165 - 0.999i)T \)
71 \( 1 + (0.809 - 0.587i)T \)
73 \( 1 + (0.724 - 0.689i)T \)
79 \( 1 + (0.999 + 0.0330i)T \)
83 \( 1 + (0.371 + 0.928i)T \)
89 \( 1 + (-0.371 - 0.928i)T \)
97 \( 1 + (-0.340 - 0.940i)T \)
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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

−23.06159958877386959099639031379, −22.22187339655817997313658082313, −21.56593190334937442367309480988, −20.48822474953916666115480062958, −20.003268656905573346018653776501, −19.63285127581797949434528593974, −17.909755197701107553637159475832, −17.46858362016041850354161709408, −16.690107806835802591619268982840, −15.86973364281432304011977835172, −15.411283564524249759662587071624, −13.91422233924465805965727196028, −12.92023398896798868814525591801, −12.08423539115524159495265573343, −11.01917556090186420106867068935, −10.16690268905865193682775633398, −9.41990435703932688965217284198, −9.04935580890560762160808485107, −7.87820336183660698469284155063, −6.65658141630275792442913996145, −5.68874253816808531617547713953, −4.402023493877555159744247551577, −3.50625654379048835474500589851, −2.34987080851331051178879541670, −0.91954570091747890744653808164, 0.15587363930632668104001057909, 1.64060328958632739224153901578, 2.37807941770641537434493787759, 3.51706018176940441039670623559, 5.74867978811924489266339663543, 6.3097936948123707559695229808, 6.82474190615884472484978241007, 7.83779249433879000633353950802, 8.98316078843996982309748713453, 9.50439719882550819473831427357, 10.75275676860977784346175855709, 11.51082567881881257334724436681, 12.33442705697285292764441481585, 13.72238090159258755664674265505, 14.169595010831997395391006036540, 15.24562750228174870280698413281, 16.329730964455952902135190086547, 16.99707917629542188571837560235, 17.98095194518378878601020508578, 18.6148182505412750991382734076, 19.12633890047419688091684877676, 19.79416321844983864390346494290, 20.944816167111540994883284369069, 22.32806257479425424693408613472, 22.59296675515213215957128448709

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