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Properties

Label 1-283-283.32-r1-0-0
Degree $1$
Conductor $283$
Sign $-0.604 - 0.796i$
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  + (0.0334 + 0.999i)2-s + (0.420 − 0.907i)3-s + (−0.997 + 0.0667i)4-s + (0.645 − 0.763i)5-s + (0.920 + 0.390i)6-s + (−0.166 − 0.986i)7-s + (−0.100 − 0.994i)8-s + (−0.645 − 0.763i)9-s + (0.784 + 0.619i)10-s + (−0.997 − 0.0667i)11-s + (−0.359 + 0.933i)12-s + (0.920 + 0.390i)13-s + (0.979 − 0.199i)14-s + (−0.420 − 0.907i)15-s + (0.991 − 0.133i)16-s + (0.979 + 0.199i)17-s + ⋯
L(s)  = 1  + (0.0334 + 0.999i)2-s + (0.420 − 0.907i)3-s + (−0.997 + 0.0667i)4-s + (0.645 − 0.763i)5-s + (0.920 + 0.390i)6-s + (−0.166 − 0.986i)7-s + (−0.100 − 0.994i)8-s + (−0.645 − 0.763i)9-s + (0.784 + 0.619i)10-s + (−0.997 − 0.0667i)11-s + (−0.359 + 0.933i)12-s + (0.920 + 0.390i)13-s + (0.979 − 0.199i)14-s + (−0.420 − 0.907i)15-s + (0.991 − 0.133i)16-s + (0.979 + 0.199i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.604 - 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.604 - 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $-0.604 - 0.796i$
Analytic conductor: \(30.4125\)
Root analytic conductor: \(30.4125\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (1:\ ),\ -0.604 - 0.796i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6097429997 - 1.228105329i\)
\(L(\frac12)\) \(\approx\) \(0.6097429997 - 1.228105329i\)
\(L(1)\) \(\approx\) \(1.040332379 - 0.2319563204i\)
\(L(1)\) \(\approx\) \(1.040332379 - 0.2319563204i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (0.0334 + 0.999i)T \)
3 \( 1 + (0.420 - 0.907i)T \)
5 \( 1 + (0.645 - 0.763i)T \)
7 \( 1 + (-0.166 - 0.986i)T \)
11 \( 1 + (-0.997 - 0.0667i)T \)
13 \( 1 + (0.920 + 0.390i)T \)
17 \( 1 + (0.979 + 0.199i)T \)
19 \( 1 + (-0.920 - 0.390i)T \)
23 \( 1 + (0.860 - 0.509i)T \)
29 \( 1 + (-0.741 - 0.670i)T \)
31 \( 1 + (-0.593 + 0.805i)T \)
37 \( 1 + (-0.480 + 0.876i)T \)
41 \( 1 + (0.100 - 0.994i)T \)
43 \( 1 + (0.538 + 0.842i)T \)
47 \( 1 + (-0.920 + 0.390i)T \)
53 \( 1 + (-0.991 - 0.133i)T \)
59 \( 1 + (-0.944 + 0.328i)T \)
61 \( 1 + (0.784 - 0.619i)T \)
67 \( 1 + (-0.359 - 0.933i)T \)
71 \( 1 + (-0.997 - 0.0667i)T \)
73 \( 1 + (0.593 + 0.805i)T \)
79 \( 1 + (0.538 - 0.842i)T \)
83 \( 1 + (0.964 - 0.264i)T \)
89 \( 1 + (-0.538 - 0.842i)T \)
97 \( 1 + (0.860 - 0.509i)T \)
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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

−25.74996640106808679596412043959, −25.29743743057322267571994199103, −23.39925785934863535901853729138, −22.642714825542750828517226470856, −21.79357298314384959767232394566, −21.08804154991076406764217236103, −20.663393554990239379047286623749, −19.18218095141009032685843781897, −18.64036884832704609186365950380, −17.76153929624183450721859443199, −16.464194040697164696063529639094, −15.19608647609799850564630111284, −14.62225314577935086526598884633, −13.491741660979428018074896746098, −12.699239437561013116739493084084, −11.215147058514524437981256831536, −10.62994951776608285993256914861, −9.72589484773727255434204244030, −8.946525331199906370412171497708, −7.88488841828978351031603715160, −5.80610400053617907011759853802, −5.2233814619674328079174136924, −3.603541275987332428332604711019, −2.890993615134830737234008116367, −1.92292782596410760063713498888, 0.38636921516048829911181002429, 1.51219554178917384064892466709, 3.32387337448903001781790869015, 4.66989292446419641573924584258, 5.86014514664053237738077484936, 6.710036750830742043453378076737, 7.775520257421108672814396119590, 8.53195358891891606734892466661, 9.48632864237857297175532547754, 10.71910406509610169890419089858, 12.57493955766284386896502223507, 13.148812068536527328826772495878, 13.774787780803511107058031781130, 14.65806900128168376768310153481, 15.970636103537255495845225094207, 16.87241317957701533545778871709, 17.51401743524145977877384109312, 18.52685581713897282467865755733, 19.292012905485260399538965203696, 20.66774287064400271226700320111, 21.19134145984551625369747109726, 22.84415731147892390046780464399, 23.62275044734765977852605545325, 24.02207400839697959450720737508, 25.08769511056921758676348936915

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