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Properties

Label 1-283-283.14-r1-0-0
Degree $1$
Conductor $283$
Sign $-0.986 + 0.162i$
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 + (−0.951 − 0.306i)3-s + (0.231 − 0.972i)4-s + (0.911 + 0.410i)5-s + (0.937 − 0.348i)6-s + (0.317 − 0.948i)7-s + (0.420 + 0.907i)8-s + (0.811 + 0.584i)9-s + (−0.970 + 0.242i)10-s + (−0.958 − 0.285i)11-s + (−0.519 + 0.854i)12-s + (−0.770 − 0.637i)13-s + (0.338 + 0.940i)14-s + (−0.741 − 0.670i)15-s + (−0.892 − 0.451i)16-s + (−0.984 + 0.177i)17-s + ⋯
L(s)  = 1  + (−0.784 + 0.619i)2-s + (−0.951 − 0.306i)3-s + (0.231 − 0.972i)4-s + (0.911 + 0.410i)5-s + (0.937 − 0.348i)6-s + (0.317 − 0.948i)7-s + (0.420 + 0.907i)8-s + (0.811 + 0.584i)9-s + (−0.970 + 0.242i)10-s + (−0.958 − 0.285i)11-s + (−0.519 + 0.854i)12-s + (−0.770 − 0.637i)13-s + (0.338 + 0.940i)14-s + (−0.741 − 0.670i)15-s + (−0.892 − 0.451i)16-s + (−0.984 + 0.177i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.986 + 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.986 + 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $-0.986 + 0.162i$
Analytic conductor: \(30.4125\)
Root analytic conductor: \(30.4125\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (1:\ ),\ -0.986 + 0.162i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.002834716100 + 0.03469128574i\)
\(L(\frac12)\) \(\approx\) \(0.002834716100 + 0.03469128574i\)
\(L(1)\) \(\approx\) \(0.5225575825 + 0.02912143406i\)
\(L(1)\) \(\approx\) \(0.5225575825 + 0.02912143406i\)

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 \( 1 + (-0.951 - 0.306i)T \)
5 \( 1 + (0.911 + 0.410i)T \)
7 \( 1 + (0.317 - 0.948i)T \)
11 \( 1 + (-0.958 - 0.285i)T \)
13 \( 1 + (-0.770 - 0.637i)T \)
17 \( 1 + (-0.984 + 0.177i)T \)
19 \( 1 + (0.166 - 0.986i)T \)
23 \( 1 + (0.975 - 0.220i)T \)
29 \( 1 + (-0.538 + 0.842i)T \)
31 \( 1 + (0.380 + 0.924i)T \)
37 \( 1 + (-0.902 + 0.431i)T \)
41 \( 1 + (0.996 - 0.0890i)T \)
43 \( 1 + (-0.359 - 0.933i)T \)
47 \( 1 + (-0.937 - 0.348i)T \)
53 \( 1 + (0.892 - 0.451i)T \)
59 \( 1 + (-0.122 + 0.992i)T \)
61 \( 1 + (0.695 - 0.718i)T \)
67 \( 1 + (-0.480 + 0.876i)T \)
71 \( 1 + (0.231 + 0.972i)T \)
73 \( 1 + (-0.380 + 0.924i)T \)
79 \( 1 + (-0.359 + 0.933i)T \)
83 \( 1 + (0.400 - 0.916i)T \)
89 \( 1 + (-0.987 - 0.155i)T \)
97 \( 1 + (-0.679 - 0.734i)T \)
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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

−24.835618766137118060593618994951, −24.35896159201117065418905218443, −22.766456750713638996394989929490, −21.990907582550452018513363640448, −21.004006445888150575601600554746, −20.923777507002030797821837107919, −19.23155239741522762988339878701, −18.28692103725156051474832624172, −17.718640122307499001153017364692, −16.90831876250713452920300829958, −16.05682318547981780329919600953, −15.02051247155590533697649417661, −13.30010660519112030038326841608, −12.50575768837644074979560748951, −11.67228666128478660022839855535, −10.73545416678635018606896143108, −9.69926788060747664275952902060, −9.17589715492563456875542238085, −7.819769000074897215331482280029, −6.50568587607360587748341856247, −5.37153951015927635791060064881, −4.44086010913345560647391579007, −2.55565484224861761920874028961, −1.64226127214891281694527595932, −0.016638623759159075897232657, 1.17843289295254140995446777655, 2.50838269212512778588203085960, 4.905372863406819772322862719049, 5.4661545291739476809239403120, 6.89398778898670758030971115633, 7.13608442551729641282278452558, 8.556867152380728695115292271000, 9.962498834575267852832525575807, 10.62150045817167005220807881185, 11.18885786923866052609332354866, 12.973422659171320952664364217228, 13.66875309098915958817645336449, 14.86810717315840336611096310810, 15.902873348691341014607485443497, 16.94787305595312755341329889627, 17.58952478810523238652346172672, 18.03158638398379090515067295797, 19.09481230608351885641604014930, 20.158523340728966018708822058032, 21.33138913417928190921561537464, 22.4110067162741424019370086907, 23.24071061082026979812119838334, 24.19591721344228076650080519996, 24.68278045263491925124646446212, 25.95606811477351891842289806066

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