Properties

Degree $1$
Conductor $283$
Sign $-0.0907 + 0.995i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.593 + 0.805i)2-s + (0.798 + 0.602i)3-s + (−0.296 − 0.955i)4-s + (0.970 − 0.242i)5-s + (−0.958 + 0.285i)6-s + (−0.848 − 0.528i)7-s + (0.944 + 0.328i)8-s + (0.274 + 0.961i)9-s + (−0.380 + 0.924i)10-s + (−0.679 − 0.734i)11-s + (0.338 − 0.940i)12-s + (0.726 + 0.687i)13-s + (0.929 − 0.369i)14-s + (0.920 + 0.390i)15-s + (−0.824 + 0.565i)16-s + (−0.144 − 0.989i)17-s + ⋯
L(s,χ)  = 1  + (−0.593 + 0.805i)2-s + (0.798 + 0.602i)3-s + (−0.296 − 0.955i)4-s + (0.970 − 0.242i)5-s + (−0.958 + 0.285i)6-s + (−0.848 − 0.528i)7-s + (0.944 + 0.328i)8-s + (0.274 + 0.961i)9-s + (−0.380 + 0.924i)10-s + (−0.679 − 0.734i)11-s + (0.338 − 0.940i)12-s + (0.726 + 0.687i)13-s + (0.929 − 0.369i)14-s + (0.920 + 0.390i)15-s + (−0.824 + 0.565i)16-s + (−0.144 − 0.989i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.0907 + 0.995i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.0907 + 0.995i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $-0.0907 + 0.995i$
Motivic weight: \(0\)
Character: $\chi_{283} (22, \cdot )$
Sato-Tate group: $\mu(282)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (1:\ ),\ -0.0907 + 0.995i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(1.321540142 + 1.447458082i\)
\(L(\frac12,\chi)\) \(\approx\) \(1.321540142 + 1.447458082i\)
\(L(\chi,1)\) \(\approx\) \(1.010978474 + 0.5567694523i\)
\(L(1,\chi)\) \(\approx\) \(1.010978474 + 0.5567694523i\)

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

−25.496349797968769928875649282337, −24.606388220266922477336918153329, −23.10459804924990779795524334740, −22.20086000101068715676303181902, −21.178969198872959327721655742329, −20.55931249770237779195420537335, −19.53847837749165709646566294855, −18.80505555928031113730091708365, −17.96818439777931313185980566785, −17.41157631617588356549806542787, −15.88273334725501656570449197585, −14.87465780313761071664624270078, −13.41662977862920567007233434671, −13.06172431143671190072997869686, −12.26375233686976369542939271890, −10.68466991116686610664774373755, −9.90345160742511629433425228797, −9.02991343547618992547376345422, −8.19535449033952922173846666694, −6.9645408374277065775876419892, −5.930154819781247417535927601693, −4.04316610244791542536282454668, −2.60985175125119007882013962194, −2.342502164825705284838873397151, −0.76173949588818165239915515879, 1.134810844001020127626085084401, 2.634100710455116245603492959852, 4.06855043999211454496844217651, 5.36584886278909522271518121896, 6.2981729660495794920514722952, 7.504149411284859602138587880168, 8.60904689928111966702363122697, 9.41301782671075822997041024641, 10.06796851450838438059618271868, 10.97399058481188628732414755810, 13.15140328962420741463334358634, 13.753575734911878388315424573474, 14.38407598357121283552621550997, 15.88759785188786038973837189030, 16.185694015497954168115823472955, 17.08398028875921344522874434256, 18.38409873348282401314828161067, 19.036690252477914862149901029316, 20.13575683174175307927220483397, 20.93927290444587328632757227564, 21.919089434696409671121572008978, 23.02210128743495891801395091437, 24.029782058281855494433719495217, 25.11265524907324548447580257974, 25.626477515223954383964637985816

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