Properties

Degree $1$
Conductor $283$
Sign $-0.931 + 0.363i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.741 + 0.670i)2-s + (−0.610 + 0.791i)3-s + (0.100 − 0.994i)4-s + (−0.711 − 0.703i)5-s + (−0.0779 − 0.996i)6-s + (−0.871 − 0.490i)7-s + (0.593 + 0.805i)8-s + (−0.253 − 0.967i)9-s + (0.999 + 0.0445i)10-s + (0.811 − 0.584i)11-s + (0.726 + 0.687i)12-s + (0.902 + 0.431i)13-s + (0.975 − 0.220i)14-s + (0.991 − 0.133i)15-s + (−0.979 − 0.199i)16-s + (−0.679 + 0.734i)17-s + ⋯
L(s,χ)  = 1  + (−0.741 + 0.670i)2-s + (−0.610 + 0.791i)3-s + (0.100 − 0.994i)4-s + (−0.711 − 0.703i)5-s + (−0.0779 − 0.996i)6-s + (−0.871 − 0.490i)7-s + (0.593 + 0.805i)8-s + (−0.253 − 0.967i)9-s + (0.999 + 0.0445i)10-s + (0.811 − 0.584i)11-s + (0.726 + 0.687i)12-s + (0.902 + 0.431i)13-s + (0.975 − 0.220i)14-s + (0.991 − 0.133i)15-s + (−0.979 − 0.199i)16-s + (−0.679 + 0.734i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.931 + 0.363i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.931 + 0.363i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $-0.931 + 0.363i$
Motivic weight: \(0\)
Character: $\chi_{283} (23, \cdot )$
Sato-Tate group: $\mu(141)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (0:\ ),\ -0.931 + 0.363i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(0.04599804222 + 0.2445598346i\)
\(L(\frac12,\chi)\) \(\approx\) \(0.04599804222 + 0.2445598346i\)
\(L(\chi,1)\) \(\approx\) \(0.3972071761 + 0.1770465550i\)
\(L(1,\chi)\) \(\approx\) \(0.3972071761 + 0.1770465550i\)

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

−25.47419166497588084221998103032, −24.38197282999694269275190098395, −23.013488072412059175611868617575, −22.526293653051245842521018139070, −21.80424022145773018666086157329, −20.12854323148123482558520689108, −19.64793551269343177927985744240, −18.62008544892034127208639758946, −18.24728461484543720836724936605, −17.181602651325341857499590016301, −16.16738195826980325115510038351, −15.30035838474230983767006186680, −13.64066336708555244635955989037, −12.7190737343608629992645428325, −11.881251375939208086405314852254, −11.21362924062503152313701264813, −10.24887398997696981658460266718, −9.01024991446765696958156858577, −7.96370854875666597642003371843, −6.87443009201602314786910983470, −6.29540178784626424180842022394, −4.30432197011463849264030022415, −3.05587023181362357487333223970, −1.97766942556895324602225117776, −0.259706647969032478222534837443, 1.21373730073310020412161629087, 3.75894557208129447136651037025, 4.42368534616268337788893655438, 6.041355629409977210990274475556, 6.436727532453937863082292190042, 8.03259525061010223809167616515, 8.93985489353627402270430809866, 9.72580210005742578019831017006, 10.852879946770838067307998347727, 11.57119141679341884883620744127, 12.921269090070708032895280003183, 14.222108794424655386832411539093, 15.408784963632255606122579607643, 16.053361065084960006180166410193, 16.7255451036396297450283517678, 17.28405431266778750148898641675, 18.678950162844275538987782988289, 19.6049461354082077933702402809, 20.266895604560355008127125230445, 21.47125703575711600726553413745, 22.68427868545851896767352263719, 23.39319795816229323512474626259, 24.04731790315248678493327460049, 25.18174070672449333676165223418, 26.30418572963478047769752278569

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