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Properties

Label 1-571-571.29-r0-0-0
Degree $1$
Conductor $571$
Sign $-0.995 + 0.0953i$
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  + (0.137 + 0.990i)2-s + (−0.592 + 0.805i)3-s + (−0.962 + 0.272i)4-s + (0.904 + 0.426i)5-s + (−0.879 − 0.475i)6-s + (0.945 + 0.324i)7-s + (−0.401 − 0.915i)8-s + (−0.298 − 0.954i)9-s + (−0.298 + 0.954i)10-s + (0.975 + 0.218i)11-s + (0.350 − 0.936i)12-s + (−0.998 − 0.0550i)13-s + (−0.191 + 0.981i)14-s + (−0.879 + 0.475i)15-s + (0.851 − 0.523i)16-s + (0.851 − 0.523i)17-s + ⋯
L(s)  = 1  + (0.137 + 0.990i)2-s + (−0.592 + 0.805i)3-s + (−0.962 + 0.272i)4-s + (0.904 + 0.426i)5-s + (−0.879 − 0.475i)6-s + (0.945 + 0.324i)7-s + (−0.401 − 0.915i)8-s + (−0.298 − 0.954i)9-s + (−0.298 + 0.954i)10-s + (0.975 + 0.218i)11-s + (0.350 − 0.936i)12-s + (−0.998 − 0.0550i)13-s + (−0.191 + 0.981i)14-s + (−0.879 + 0.475i)15-s + (0.851 − 0.523i)16-s + (0.851 − 0.523i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.995 + 0.0953i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ -0.995 + 0.0953i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06373666432 + 1.333915792i\)
\(L(\frac12)\) \(\approx\) \(0.06373666432 + 1.333915792i\)
\(L(1)\) \(\approx\) \(0.6438195362 + 0.8671267699i\)
\(L(1)\) \(\approx\) \(0.6438195362 + 0.8671267699i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (0.137 + 0.990i)T \)
3 \( 1 + (-0.592 + 0.805i)T \)
5 \( 1 + (0.904 + 0.426i)T \)
7 \( 1 + (0.945 + 0.324i)T \)
11 \( 1 + (0.975 + 0.218i)T \)
13 \( 1 + (-0.998 - 0.0550i)T \)
17 \( 1 + (0.851 - 0.523i)T \)
19 \( 1 + (-0.592 + 0.805i)T \)
23 \( 1 + (-0.401 + 0.915i)T \)
29 \( 1 + (0.451 + 0.892i)T \)
31 \( 1 + (-0.879 - 0.475i)T \)
37 \( 1 + (-0.998 + 0.0550i)T \)
41 \( 1 + (0.904 + 0.426i)T \)
43 \( 1 + (-0.298 + 0.954i)T \)
47 \( 1 + (0.0275 + 0.999i)T \)
53 \( 1 + (0.137 - 0.990i)T \)
59 \( 1 + (0.945 + 0.324i)T \)
61 \( 1 + (-0.926 - 0.376i)T \)
67 \( 1 + (-0.926 + 0.376i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (0.451 - 0.892i)T \)
79 \( 1 + (0.716 - 0.697i)T \)
83 \( 1 + (0.0275 + 0.999i)T \)
89 \( 1 + (0.851 - 0.523i)T \)
97 \( 1 + (-0.962 + 0.272i)T \)
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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

−22.69000825388018238735470360361, −21.84229141374896475190745642777, −21.36825767585741264824668535831, −20.29344883836175720589560209575, −19.55600498278089352911915360430, −18.73506956926176009405137701568, −17.74805170523219908030331945061, −17.254857727271125502943134133822, −16.73675066730878878139164388522, −14.70148085081830648240572836244, −14.11531573224001095426852077444, −13.39546999153021130644644697511, −12.320574786820024222902831019757, −12.00036394153588869772058420903, −10.832513967032691646448344782963, −10.2236049493092584675727808257, −9.016507800043159049092672294010, −8.22885730843398370501326536213, −6.95343365627835637297942411302, −5.80858128859696675469587656678, −5.043225039109392901254611284747, −4.15376186161619587387229565033, −2.42705017303038759776690083690, −1.73111692057529899302135104296, −0.77529731963234409508199607018, 1.52787942990234451029055257561, 3.225883181434419933145884047490, 4.38659281322686453140975663401, 5.24928609100961841281736585816, 5.85745988709074746427848688015, 6.82214774061146184231521579034, 7.844527949655074095658182129967, 9.14955054931743134719268962154, 9.63485299047152706444341079774, 10.5811766949183424280217758033, 11.790341533921499395038345688049, 12.515656701333445451763717643074, 13.97920220579073331324106624490, 14.6860701904006820950214250312, 14.88862246686417165477195447571, 16.281663050784981477999728268420, 16.89312537894078665404098607926, 17.69846871687098005171933461746, 18.04623798892596137013238801229, 19.29720744089857229030485934543, 20.79935097328540547564404853652, 21.43990266875665462143143949987, 22.121491716052447931113062762700, 22.680639302538207247035186577152, 23.65198386726868698652879545430

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