Properties

Degree $1$
Conductor $283$
Sign $-0.143 + 0.989i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.944 − 0.328i)2-s + (0.628 + 0.777i)3-s + (0.784 + 0.619i)4-s + (0.951 + 0.306i)5-s + (−0.338 − 0.940i)6-s + (−0.911 + 0.410i)7-s + (−0.538 − 0.842i)8-s + (−0.210 + 0.977i)9-s + (−0.798 − 0.602i)10-s + (−0.929 − 0.369i)11-s + (0.0111 + 0.999i)12-s + (0.984 + 0.177i)13-s + (0.996 − 0.0890i)14-s + (0.359 + 0.933i)15-s + (0.231 + 0.972i)16-s + (−0.575 + 0.818i)17-s + ⋯
L(s,χ)  = 1  + (−0.944 − 0.328i)2-s + (0.628 + 0.777i)3-s + (0.784 + 0.619i)4-s + (0.951 + 0.306i)5-s + (−0.338 − 0.940i)6-s + (−0.911 + 0.410i)7-s + (−0.538 − 0.842i)8-s + (−0.210 + 0.977i)9-s + (−0.798 − 0.602i)10-s + (−0.929 − 0.369i)11-s + (0.0111 + 0.999i)12-s + (0.984 + 0.177i)13-s + (0.996 − 0.0890i)14-s + (0.359 + 0.933i)15-s + (0.231 + 0.972i)16-s + (−0.575 + 0.818i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.143 + 0.989i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.143 + 0.989i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $-0.143 + 0.989i$
Motivic weight: \(0\)
Character: $\chi_{283} (41, \cdot )$
Sato-Tate group: $\mu(141)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (0:\ ),\ -0.143 + 0.989i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(0.6089631949 + 0.7034551073i\)
\(L(\frac12,\chi)\) \(\approx\) \(0.6089631949 + 0.7034551073i\)
\(L(\chi,1)\) \(\approx\) \(0.7991659342 + 0.3247552517i\)
\(L(1,\chi)\) \(\approx\) \(0.7991659342 + 0.3247552517i\)

Euler product

   \(L(\chi,s) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)
   \(L(s,\chi) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−25.3553030268293618451954621259, −24.8822203893684551846244533182, −23.71780197368200337324083044547, −23.0447613255283003599475345660, −21.34999725454213366592902512229, −20.391879059124540687178459328547, −19.91241395670488614575852522951, −18.68514088041699044540722622488, −18.138170471372332583025650090, −17.31705824778561981662761066363, −16.21315923115714480097940318605, −15.382042235159226904844577803437, −14.07996004691946215414899772564, −13.28848647000037500046184482943, −12.502375062897628299984757907386, −10.85879738326338220367202023442, −9.98640610772140986389347233199, −8.983393801696807644732455815640, −8.36166682114144153901013014013, −6.8590351201619892467411023717, −6.58586944517248802234226131136, −5.16884425090596102450943861022, −3.02273581843712139614100627127, −2.114774038489035287619729119780, −0.754590086828916327482656271025, 1.895190530191411905162594366319, 2.836111554209765498944118404864, 3.77249795192425873599435798499, 5.681254025907738999142229448171, 6.56781422260556115514477440785, 8.15221870668227235704414264304, 8.8361908264573747497965058537, 9.86960629028531119442018584644, 10.35180338168910174431108457421, 11.33457394689865285053093635941, 12.99222948158245187397784951782, 13.53931995756835840679038207929, 15.09974406355655650458858748390, 15.77192224724973496405789457622, 16.66297809543842730410633970233, 17.61858332259327208259651968656, 18.85817084872944279721887355045, 19.166654084145460275960292298181, 20.52314210491373152356102291645, 21.20738935242474088284158811782, 21.73664910087510797552359009029, 22.83066377686309578300110574352, 24.45700900174249952148096316694, 25.64786299502227883873970631236, 25.73948856115319829463177784903

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