Properties

Degree 1
Conductor 569
Sign $0.814 - 0.580i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(569\)
\( \varepsilon \)  =  $0.814 - 0.580i$
motivic weight  =  \(0\)
character  :  $\chi_{569} (10, \cdot )$
Sato-Tate  :  $\mu(284)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 569,\ (0:\ ),\ 0.814 - 0.580i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.159577067 - 0.3711017574i$
$L(\frac12,\chi)$  $\approx$  $1.159577067 - 0.3711017574i$
$L(\chi,1)$  $\approx$  0.9243916291 - 0.2306470444i
$L(1,\chi)$  $\approx$  0.9243916291 - 0.2306470444i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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