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Properties

Label 1-283-283.17-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: $\chi_{283} (17, \cdot )$
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

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

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