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Properties

Label 1-283-283.46-r1-0-0
Degree $1$
Conductor $283$
Sign $0.960 + 0.276i$
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  + (0.979 + 0.199i)2-s + (0.871 − 0.490i)3-s + (0.920 + 0.390i)4-s + (0.999 − 0.0222i)5-s + (0.951 − 0.306i)6-s + (−0.460 + 0.887i)7-s + (0.824 + 0.565i)8-s + (0.519 − 0.854i)9-s + (0.984 + 0.177i)10-s + (−0.798 − 0.602i)11-s + (0.993 − 0.111i)12-s + (−0.210 + 0.977i)13-s + (−0.628 + 0.777i)14-s + (0.860 − 0.509i)15-s + (0.695 + 0.718i)16-s + (0.987 − 0.155i)17-s + ⋯
L(s)  = 1  + (0.979 + 0.199i)2-s + (0.871 − 0.490i)3-s + (0.920 + 0.390i)4-s + (0.999 − 0.0222i)5-s + (0.951 − 0.306i)6-s + (−0.460 + 0.887i)7-s + (0.824 + 0.565i)8-s + (0.519 − 0.854i)9-s + (0.984 + 0.177i)10-s + (−0.798 − 0.602i)11-s + (0.993 − 0.111i)12-s + (−0.210 + 0.977i)13-s + (−0.628 + 0.777i)14-s + (0.860 − 0.509i)15-s + (0.695 + 0.718i)16-s + (0.987 − 0.155i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.960 + 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.960 + 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $0.960 + 0.276i$
Analytic conductor: \(30.4125\)
Root analytic conductor: \(30.4125\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (1:\ ),\ 0.960 + 0.276i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.864450369 + 0.8273263075i\)
\(L(\frac12)\) \(\approx\) \(5.864450369 + 0.8273263075i\)
\(L(1)\) \(\approx\) \(2.936740759 + 0.2298647834i\)
\(L(1)\) \(\approx\) \(2.936740759 + 0.2298647834i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (0.979 + 0.199i)T \)
3 \( 1 + (0.871 - 0.490i)T \)
5 \( 1 + (0.999 - 0.0222i)T \)
7 \( 1 + (-0.460 + 0.887i)T \)
11 \( 1 + (-0.798 - 0.602i)T \)
13 \( 1 + (-0.210 + 0.977i)T \)
17 \( 1 + (0.987 - 0.155i)T \)
19 \( 1 + (0.741 + 0.670i)T \)
23 \( 1 + (0.556 - 0.830i)T \)
29 \( 1 + (-0.296 + 0.955i)T \)
31 \( 1 + (-0.144 - 0.989i)T \)
37 \( 1 + (0.380 + 0.924i)T \)
41 \( 1 + (-0.0779 - 0.996i)T \)
43 \( 1 + (-0.964 - 0.264i)T \)
47 \( 1 + (-0.951 - 0.306i)T \)
53 \( 1 + (-0.695 + 0.718i)T \)
59 \( 1 + (0.996 - 0.0890i)T \)
61 \( 1 + (-0.645 - 0.763i)T \)
67 \( 1 + (-0.593 + 0.805i)T \)
71 \( 1 + (0.920 - 0.390i)T \)
73 \( 1 + (0.144 - 0.989i)T \)
79 \( 1 + (-0.964 + 0.264i)T \)
83 \( 1 + (-0.848 - 0.528i)T \)
89 \( 1 + (-0.253 - 0.967i)T \)
97 \( 1 + (0.441 + 0.897i)T \)
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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

−25.38460916786417037531033106893, −24.53073235041755911825722001954, −23.28617311946834758680619880826, −22.56487862160629650518971870702, −21.49648618071715031675108076481, −20.92607949631581130815939595063, −20.120186572190102816254794774636, −19.43407579049555268961494124252, −18.01030468367445693348147483319, −16.78475141866704310585625688032, −15.83169823816214663775874837086, −14.92870466514928259707932726874, −14.08643308766611525366647535758, −13.21979949098219654082346092832, −12.82725211411798671130810806800, −11.036531893114092234082149379190, −10.00417583572038120894104656363, −9.75539106844356744045192788603, −7.85130181616929971425302391740, −7.02420046337941619646651636400, −5.54666198826229624530504861198, −4.781926543616244415892297863143, −3.38360019154377275495954677090, −2.726066324260928653430087462917, −1.360452090906633070968755968560, 1.61793509456385972593036491303, 2.62813765970526273315504274936, 3.37707309836375807681170215783, 5.07757153550332429637455970713, 5.9899113107240111423552518351, 6.88911155562299106998915284264, 8.08249818897826553516502285249, 9.156963709653924544296953276137, 10.19108619539799518771989881282, 11.74740404791110881311342641750, 12.65563398743745325791384646086, 13.36333419265504083665555179538, 14.212468561513643512809595734984, 14.85330068566809170210475164520, 16.05245418068192299481733374964, 16.846107099605740135702324324934, 18.50589703941336896425557456744, 18.81028966283065103949317379423, 20.28501983973032663332960684229, 21.050846381756282192504007022819, 21.63882756973707316169019493498, 22.59775798299949195460314655690, 23.87099037637415273415984143641, 24.48562360974394006144953749309, 25.34250242813717464356894071242

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