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Properties

Label 1-283-283.22-r1-0-0
Degree $1$
Conductor $283$
Sign $-0.0907 + 0.995i$
Analytic cond. $30.4125$
Root an. cond. $30.4125$
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  + (−0.593 + 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $-0.0907 + 0.995i$
Analytic conductor: \(30.4125\)
Root analytic conductor: \(30.4125\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (1:\ ),\ -0.0907 + 0.995i)\)

Particular Values

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

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (-0.593 + 0.805i)T \)
3 \( 1 + (0.798 + 0.602i)T \)
5 \( 1 + (0.970 - 0.242i)T \)
7 \( 1 + (-0.848 - 0.528i)T \)
11 \( 1 + (-0.679 - 0.734i)T \)
13 \( 1 + (0.726 + 0.687i)T \)
17 \( 1 + (-0.144 - 0.989i)T \)
19 \( 1 + (-0.231 + 0.972i)T \)
23 \( 1 + (-0.210 + 0.977i)T \)
29 \( 1 + (-0.166 + 0.986i)T \)
31 \( 1 + (0.999 - 0.0222i)T \)
37 \( 1 + (0.911 + 0.410i)T \)
41 \( 1 + (0.756 + 0.654i)T \)
43 \( 1 + (0.979 - 0.199i)T \)
47 \( 1 + (0.958 + 0.285i)T \)
53 \( 1 + (0.824 + 0.565i)T \)
59 \( 1 + (0.556 - 0.830i)T \)
61 \( 1 + (0.991 + 0.133i)T \)
67 \( 1 + (0.645 - 0.763i)T \)
71 \( 1 + (-0.296 + 0.955i)T \)
73 \( 1 + (-0.999 - 0.0222i)T \)
79 \( 1 + (0.979 + 0.199i)T \)
83 \( 1 + (-0.987 + 0.155i)T \)
89 \( 1 + (0.317 - 0.948i)T \)
97 \( 1 + (0.951 - 0.306i)T \)
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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

−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