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Properties

Label 1-569-569.57-r0-0-0
Degree $1$
Conductor $569$
Sign $0.814 + 0.580i$
Analytic cond. $2.64242$
Root an. cond. $2.64242$
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  + (−0.325 + 0.945i)2-s + (−0.873 − 0.487i)3-s + (−0.787 − 0.616i)4-s + (0.984 − 0.176i)5-s + (0.745 − 0.666i)6-s + (0.991 − 0.132i)7-s + (0.839 − 0.544i)8-s + (0.525 + 0.850i)9-s + (−0.154 + 0.988i)10-s + (0.801 − 0.598i)11-s + (0.387 + 0.921i)12-s + (0.883 + 0.467i)13-s + (−0.197 + 0.980i)14-s + (−0.945 − 0.325i)15-s + (0.240 + 0.970i)16-s + (−0.903 + 0.428i)17-s + ⋯
L(s)  = 1  + (−0.325 + 0.945i)2-s + (−0.873 − 0.487i)3-s + (−0.787 − 0.616i)4-s + (0.984 − 0.176i)5-s + (0.745 − 0.666i)6-s + (0.991 − 0.132i)7-s + (0.839 − 0.544i)8-s + (0.525 + 0.850i)9-s + (−0.154 + 0.988i)10-s + (0.801 − 0.598i)11-s + (0.387 + 0.921i)12-s + (0.883 + 0.467i)13-s + (−0.197 + 0.980i)14-s + (−0.945 − 0.325i)15-s + (0.240 + 0.970i)16-s + (−0.903 + 0.428i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.814 + 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.814 + 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $0.814 + 0.580i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ 0.814 + 0.580i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.159577067 + 0.3711017574i\)
\(L(\frac12)\) \(\approx\) \(1.159577067 + 0.3711017574i\)
\(L(1)\) \(\approx\) \(0.9243916291 + 0.2306470444i\)
\(L(1)\) \(\approx\) \(0.9243916291 + 0.2306470444i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (-0.325 + 0.945i)T \)
3 \( 1 + (-0.873 - 0.487i)T \)
5 \( 1 + (0.984 - 0.176i)T \)
7 \( 1 + (0.991 - 0.132i)T \)
11 \( 1 + (0.801 - 0.598i)T \)
13 \( 1 + (0.883 + 0.467i)T \)
17 \( 1 + (-0.903 + 0.428i)T \)
19 \( 1 + (0.616 + 0.787i)T \)
23 \( 1 + (0.930 - 0.367i)T \)
29 \( 1 + (-0.428 + 0.903i)T \)
31 \( 1 + (-0.912 + 0.408i)T \)
37 \( 1 + (-0.132 - 0.991i)T \)
41 \( 1 + (-0.367 - 0.930i)T \)
43 \( 1 + (0.325 + 0.945i)T \)
47 \( 1 + (0.219 - 0.975i)T \)
53 \( 1 + (-0.745 + 0.666i)T \)
59 \( 1 + (-0.219 + 0.975i)T \)
61 \( 1 + (-0.862 + 0.506i)T \)
67 \( 1 + (-0.903 - 0.428i)T \)
71 \( 1 + (0.839 + 0.544i)T \)
73 \( 1 + (-0.616 - 0.787i)T \)
79 \( 1 + (0.110 + 0.993i)T \)
83 \( 1 + (0.262 + 0.964i)T \)
89 \( 1 + (0.912 + 0.408i)T \)
97 \( 1 + (0.346 + 0.937i)T \)
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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

−22.7175656127619223470719089318, −22.210727268650901651614115809053, −21.53889746229145757995726814605, −20.64243925499926973605850731286, −20.2996968545715078893631645586, −18.7239399377956502324880458334, −18.039088299751337914760002403249, −17.428730995528083875964461882110, −17.02463740190711377057197172805, −15.63268296552322304423728075955, −14.66844597923942004610014959157, −13.58488281666277064940503582535, −12.87453694850570064473049986347, −11.60421714108543403400169699701, −11.24689265646953976095860968025, −10.41869142118779913607408860065, −9.39601597853871772641302529092, −8.96145646778098613636235493355, −7.45913464710849737955514062072, −6.29969516037017442174176596920, −5.146092037924127649483661594029, −4.54526049330189436488308632926, −3.29756924316930807506294243779, −1.94542269420458924047544509636, −1.0742957086059682447602946594, 1.183984382070165668972212153325, 1.7188070766764271206267013694, 4.036516207011306230260656619, 5.105445766648647355043054355479, 5.803079558867380378383782629003, 6.55421503520789038834694559308, 7.39446751298314061570841267728, 8.6239181373072331314987081961, 9.16826683126497252032745237803, 10.64703283547449479459841742126, 11.01095662800464053103184522506, 12.37342243639844066102470614041, 13.43147488538517378018420416474, 13.99212464511872752668862141429, 14.81336733305572981797654663112, 16.18344210821031566218740351489, 16.74489460815891410731987544215, 17.412331214016089080807160915889, 18.17769376821793252765265428371, 18.6069362038543812710877241442, 19.821283114730575624646891040607, 21.09943134735416916918537767313, 21.89166840392521432384097886806, 22.64326321961896428671408916722, 23.59972252287307944868517924883

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