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Properties

Label 1-283-283.50-r1-0-0
Degree $1$
Conductor $283$
Sign $0.918 + 0.396i$
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  + (−0.784 + 0.619i)2-s + (0.210 + 0.977i)3-s + (0.231 − 0.972i)4-s + (−0.811 + 0.584i)5-s + (−0.770 − 0.637i)6-s + (0.662 + 0.749i)7-s + (0.420 + 0.907i)8-s + (−0.911 + 0.410i)9-s + (0.274 − 0.961i)10-s + (0.726 − 0.687i)11-s + (0.999 + 0.0222i)12-s + (0.937 − 0.348i)13-s + (−0.984 − 0.177i)14-s + (−0.741 − 0.670i)15-s + (−0.892 − 0.451i)16-s + (0.338 − 0.940i)17-s + ⋯
L(s)  = 1  + (−0.784 + 0.619i)2-s + (0.210 + 0.977i)3-s + (0.231 − 0.972i)4-s + (−0.811 + 0.584i)5-s + (−0.770 − 0.637i)6-s + (0.662 + 0.749i)7-s + (0.420 + 0.907i)8-s + (−0.911 + 0.410i)9-s + (0.274 − 0.961i)10-s + (0.726 − 0.687i)11-s + (0.999 + 0.0222i)12-s + (0.937 − 0.348i)13-s + (−0.984 − 0.177i)14-s + (−0.741 − 0.670i)15-s + (−0.892 − 0.451i)16-s + (0.338 − 0.940i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.918 + 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.918 + 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $0.918 + 0.396i$
Analytic conductor: \(30.4125\)
Root analytic conductor: \(30.4125\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (1:\ ),\ 0.918 + 0.396i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.068026803 + 0.2205009567i\)
\(L(\frac12)\) \(\approx\) \(1.068026803 + 0.2205009567i\)
\(L(1)\) \(\approx\) \(0.6920483102 + 0.3475994991i\)
\(L(1)\) \(\approx\) \(0.6920483102 + 0.3475994991i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (-0.784 + 0.619i)T \)
3 \( 1 + (0.210 + 0.977i)T \)
5 \( 1 + (-0.811 + 0.584i)T \)
7 \( 1 + (0.662 + 0.749i)T \)
11 \( 1 + (0.726 - 0.687i)T \)
13 \( 1 + (0.937 - 0.348i)T \)
17 \( 1 + (0.338 - 0.940i)T \)
19 \( 1 + (0.166 - 0.986i)T \)
23 \( 1 + (-0.679 - 0.734i)T \)
29 \( 1 + (-0.538 + 0.842i)T \)
31 \( 1 + (0.610 - 0.791i)T \)
37 \( 1 + (0.0779 - 0.996i)T \)
41 \( 1 + (-0.575 - 0.818i)T \)
43 \( 1 + (-0.359 - 0.933i)T \)
47 \( 1 + (0.770 - 0.637i)T \)
53 \( 1 + (0.892 - 0.451i)T \)
59 \( 1 + (-0.798 - 0.602i)T \)
61 \( 1 + (0.695 - 0.718i)T \)
67 \( 1 + (-0.480 + 0.876i)T \)
71 \( 1 + (0.231 + 0.972i)T \)
73 \( 1 + (-0.610 - 0.791i)T \)
79 \( 1 + (-0.359 + 0.933i)T \)
83 \( 1 + (-0.993 + 0.111i)T \)
89 \( 1 + (0.628 - 0.777i)T \)
97 \( 1 + (0.975 - 0.220i)T \)
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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

−25.36497965983921952770455911374, −24.47866588411910480296609118999, −23.53993797262050454145147190642, −22.83981211146707235179711525853, −21.21525973383590931503467340625, −20.40756789934579339814943296849, −19.81509147855521695516744989545, −19.03920793519630417474630708202, −18.09668873274008454893077726733, −17.18513924223475736527580801181, −16.56277558778691307702343613507, −15.14880099383894219180658666520, −13.89284376175289803065116058513, −12.9434009047088303986439597844, −11.91887025105750274828051451235, −11.52006932616992247613764039842, −10.20080410219277259083096504171, −8.892399914157431829801573155036, −8.04459709917987909808023776809, −7.52249769343320787673385111303, −6.29059203482071505787252165292, −4.284256789453002028003174498524, −3.48436227800026652008284971441, −1.58344218481446269744864998758, −1.20616174370467281217137553427, 0.491452263518627522742045543249, 2.46405702305963152214551345147, 3.76503513072120933178405099849, 5.07110607559186774320949190953, 6.062346610746620438759676538966, 7.36666618908744042435390452683, 8.53744028139273108206664254468, 8.90834071497052361973939145474, 10.25519066285726610917548831085, 11.20722937026468832204432036442, 11.6777373935969009401301400705, 13.94221176497461879856373560442, 14.587105668683014264805971079880, 15.53874022730849018555499149723, 15.95459121949704086647124962884, 17.0091986793269691503181507533, 18.20987591447314629776734013064, 18.8371281774314123683309845965, 19.93731430443721588788315158854, 20.63887420396793206890867845396, 21.94595140258706994448994391772, 22.66606615201450348063738005560, 23.75652984405730925062972899922, 24.70234114824610264135066547712, 25.603869504476180578771609271339

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