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Properties

Label 1-571-571.163-r0-0-0
Degree $1$
Conductor $571$
Sign $0.900 + 0.434i$
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  + (−0.677 − 0.735i)2-s + (0.828 + 0.560i)3-s + (−0.0825 + 0.996i)4-s + (0.991 − 0.131i)5-s + (−0.148 − 0.988i)6-s + (0.746 − 0.665i)7-s + (0.789 − 0.614i)8-s + (0.371 + 0.928i)9-s + (−0.768 − 0.639i)10-s + (−0.768 + 0.639i)11-s + (−0.627 + 0.778i)12-s + (−0.574 + 0.818i)13-s + (−0.995 − 0.0990i)14-s + (0.894 + 0.446i)15-s + (−0.986 − 0.164i)16-s + (0.701 + 0.712i)17-s + ⋯
L(s)  = 1  + (−0.677 − 0.735i)2-s + (0.828 + 0.560i)3-s + (−0.0825 + 0.996i)4-s + (0.991 − 0.131i)5-s + (−0.148 − 0.988i)6-s + (0.746 − 0.665i)7-s + (0.789 − 0.614i)8-s + (0.371 + 0.928i)9-s + (−0.768 − 0.639i)10-s + (−0.768 + 0.639i)11-s + (−0.627 + 0.778i)12-s + (−0.574 + 0.818i)13-s + (−0.995 − 0.0990i)14-s + (0.894 + 0.446i)15-s + (−0.986 − 0.164i)16-s + (0.701 + 0.712i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.900 + 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.900 + 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.900 + 0.434i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ 0.900 + 0.434i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.537427333 + 0.3511611775i\)
\(L(\frac12)\) \(\approx\) \(1.537427333 + 0.3511611775i\)
\(L(1)\) \(\approx\) \(1.200245119 + 0.01925862599i\)
\(L(1)\) \(\approx\) \(1.200245119 + 0.01925862599i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (-0.677 - 0.735i)T \)
3 \( 1 + (0.828 + 0.560i)T \)
5 \( 1 + (0.991 - 0.131i)T \)
7 \( 1 + (0.746 - 0.665i)T \)
11 \( 1 + (-0.768 + 0.639i)T \)
13 \( 1 + (-0.574 + 0.818i)T \)
17 \( 1 + (0.701 + 0.712i)T \)
19 \( 1 + (-0.277 + 0.960i)T \)
23 \( 1 + (-0.340 + 0.940i)T \)
29 \( 1 + (0.945 - 0.324i)T \)
31 \( 1 + (-0.986 - 0.164i)T \)
37 \( 1 + (-0.956 + 0.293i)T \)
41 \( 1 + (-0.879 - 0.475i)T \)
43 \( 1 + (0.997 + 0.0660i)T \)
47 \( 1 + (-0.986 - 0.164i)T \)
53 \( 1 + (0.980 - 0.197i)T \)
59 \( 1 + (-0.401 - 0.915i)T \)
61 \( 1 + (0.490 - 0.871i)T \)
67 \( 1 + (0.980 - 0.197i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (-0.956 + 0.293i)T \)
79 \( 1 + (0.922 - 0.386i)T \)
83 \( 1 + (-0.148 - 0.988i)T \)
89 \( 1 + (-0.148 - 0.988i)T \)
97 \( 1 + (-0.518 - 0.854i)T \)
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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

−23.64750301062822540580588308293, −22.39585776442626305882111583243, −21.33558961453593927368372106646, −20.63190589393006518291201786024, −19.66979066283634470880316830712, −18.72624430548657347324040707658, −18.05403525808856299764175790287, −17.74201486666952873609778006553, −16.52529984679853421817603282664, −15.47890237291526230013030907557, −14.703056127068230231699059437112, −14.09178699993725229186081335004, −13.299466708639132828325853786458, −12.25555087231736877907719464036, −10.835542232440880373120019988015, −10.03369287118633706453123519321, −9.03585199776092691424311708432, −8.439169641960932749287092522839, −7.58944460870371548122470221954, −6.673897154169582915678898369880, −5.60820098875613672099271422566, −4.956740008233065143353454292019, −2.83918001466208913225877956242, −2.2052488239545883028740416727, −0.94686132040325770627702296126, 1.70925987739259305019054405489, 2.01929530788979267908498365381, 3.39533236771533202113031820456, 4.3594057707756424895212655706, 5.28710182857267905339686089265, 7.06035267903345024186647938567, 7.91666617251769628543752812557, 8.65532630777018038896864638414, 9.87236308929525475266011974921, 10.02761416941856270201565300139, 10.936323223360553389139516817262, 12.20992640844787938605884184592, 13.13152032971212111849745235666, 13.992142368454591079030395892264, 14.609245800899195123675287453, 15.911044545830593985864732151218, 16.90400245427375767295331185749, 17.39527450914520587171781889650, 18.41413497071088732687531402634, 19.22779378477925973822902247209, 20.17650661053621603675389073580, 20.828785025076873206181575985319, 21.33404924037418707822692989475, 21.90954421163403802814958277711, 23.21931652440113437818316025232

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