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Properties

Label 1-571-571.256-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: $\chi_{571} (256, \cdot )$
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

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

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