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Properties

Label 1-283-283.156-r1-0-0
Degree $1$
Conductor $283$
Sign $0.242 + 0.970i$
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  + (0.892 + 0.451i)2-s + (0.979 + 0.199i)3-s + (0.593 + 0.805i)4-s + (−0.920 + 0.390i)5-s + (0.784 + 0.619i)6-s + (0.695 − 0.718i)7-s + (0.166 + 0.986i)8-s + (0.920 + 0.390i)9-s + (−0.997 − 0.0667i)10-s + (0.593 − 0.805i)11-s + (0.420 + 0.907i)12-s + (0.784 + 0.619i)13-s + (0.944 − 0.328i)14-s + (−0.979 + 0.199i)15-s + (−0.296 + 0.955i)16-s + (0.944 + 0.328i)17-s + ⋯
L(s)  = 1  + (0.892 + 0.451i)2-s + (0.979 + 0.199i)3-s + (0.593 + 0.805i)4-s + (−0.920 + 0.390i)5-s + (0.784 + 0.619i)6-s + (0.695 − 0.718i)7-s + (0.166 + 0.986i)8-s + (0.920 + 0.390i)9-s + (−0.997 − 0.0667i)10-s + (0.593 − 0.805i)11-s + (0.420 + 0.907i)12-s + (0.784 + 0.619i)13-s + (0.944 − 0.328i)14-s + (−0.979 + 0.199i)15-s + (−0.296 + 0.955i)16-s + (0.944 + 0.328i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.242 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.242 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $0.242 + 0.970i$
Analytic conductor: \(30.4125\)
Root analytic conductor: \(30.4125\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (156, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (1:\ ),\ 0.242 + 0.970i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.863392248 + 3.015808655i\)
\(L(\frac12)\) \(\approx\) \(3.863392248 + 3.015808655i\)
\(L(1)\) \(\approx\) \(2.283206291 + 1.042920250i\)
\(L(1)\) \(\approx\) \(2.283206291 + 1.042920250i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (0.892 + 0.451i)T \)
3 \( 1 + (0.979 + 0.199i)T \)
5 \( 1 + (-0.920 + 0.390i)T \)
7 \( 1 + (0.695 - 0.718i)T \)
11 \( 1 + (0.593 - 0.805i)T \)
13 \( 1 + (0.784 + 0.619i)T \)
17 \( 1 + (0.944 + 0.328i)T \)
19 \( 1 + (-0.784 - 0.619i)T \)
23 \( 1 + (0.359 + 0.933i)T \)
29 \( 1 + (-0.645 + 0.763i)T \)
31 \( 1 + (-0.860 - 0.509i)T \)
37 \( 1 + (0.741 - 0.670i)T \)
41 \( 1 + (-0.166 + 0.986i)T \)
43 \( 1 + (-0.100 + 0.994i)T \)
47 \( 1 + (-0.784 + 0.619i)T \)
53 \( 1 + (0.296 + 0.955i)T \)
59 \( 1 + (-0.0334 - 0.999i)T \)
61 \( 1 + (-0.997 + 0.0667i)T \)
67 \( 1 + (0.420 - 0.907i)T \)
71 \( 1 + (0.593 - 0.805i)T \)
73 \( 1 + (0.860 - 0.509i)T \)
79 \( 1 + (-0.100 - 0.994i)T \)
83 \( 1 + (-0.824 - 0.565i)T \)
89 \( 1 + (0.100 - 0.994i)T \)
97 \( 1 + (0.359 + 0.933i)T \)
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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

−25.022571717388655874026456446462, −24.258305456937919510260957158038, −23.34838156698714890735695661003, −22.560786468327293663523544274514, −21.21656354418429269311296654554, −20.63883024676592913978874246848, −20.01252632472244338197505727748, −18.96022380059328942167919636946, −18.37654339845153721936975041949, −16.589597710300109454052968071878, −15.34227910551020539928029263520, −14.98008497589552709632845370450, −14.13968929525152209461519868014, −12.81594519871451883285477337029, −12.33040567838508643026440327438, −11.37040028700108010683262983143, −10.11562156371255606721841167889, −8.860921423719548018993176467284, −7.985638531986459871677639316682, −6.88928315264242556086214786515, −5.431729510593562681133530693438, −4.27594734728452854400040861543, −3.50651096356751117505897306601, −2.22313460959537800380787790174, −1.13931665555465855619765072311, 1.55389358255667092046355750406, 3.238224502655330904547417713647, 3.810238434439817830393454374442, 4.69337839418063059386273487881, 6.33239581157924162220626707507, 7.44745996862840270759588533686, 8.04075856096182204771483541046, 9.07109371066875077426773356334, 10.92392716411634472382170182283, 11.34466909723652258246947000817, 12.78868569277095849232322118362, 13.74003355464562220712045484076, 14.52602652692189920296340577493, 15.037447163737750360598380325356, 16.187576401923316303922195943594, 16.821114777042853033479178957001, 18.394425704698744359470346861284, 19.49142430989347910898661246522, 20.1419887422430361981275286190, 21.25667810378006127059101513039, 21.70238394084868584994912008719, 23.121188980348934408142931290944, 23.75946092679133790311893831644, 24.4207450757388929605837525383, 25.63267476686123078287245064974

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