Properties

Label 1-283-283.49-r0-0-0
Degree $1$
Conductor $283$
Sign $0.990 + 0.137i$
Analytic cond. $1.31424$
Root an. cond. $1.31424$
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  + (−0.997 + 0.0667i)2-s + (0.984 − 0.177i)3-s + (0.991 − 0.133i)4-s + (−0.770 + 0.637i)5-s + (−0.970 + 0.242i)6-s + (0.756 + 0.654i)7-s + (−0.979 + 0.199i)8-s + (0.937 − 0.348i)9-s + (0.726 − 0.687i)10-s + (−0.380 − 0.924i)11-s + (0.951 − 0.306i)12-s + (0.274 − 0.961i)13-s + (−0.798 − 0.602i)14-s + (−0.645 + 0.763i)15-s + (0.964 − 0.264i)16-s + (−0.122 − 0.992i)17-s + ⋯
L(s)  = 1  + (−0.997 + 0.0667i)2-s + (0.984 − 0.177i)3-s + (0.991 − 0.133i)4-s + (−0.770 + 0.637i)5-s + (−0.970 + 0.242i)6-s + (0.756 + 0.654i)7-s + (−0.979 + 0.199i)8-s + (0.937 − 0.348i)9-s + (0.726 − 0.687i)10-s + (−0.380 − 0.924i)11-s + (0.951 − 0.306i)12-s + (0.274 − 0.961i)13-s + (−0.798 − 0.602i)14-s + (−0.645 + 0.763i)15-s + (0.964 − 0.264i)16-s + (−0.122 − 0.992i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.137i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.137i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $0.990 + 0.137i$
Analytic conductor: \(1.31424\)
Root analytic conductor: \(1.31424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (0:\ ),\ 0.990 + 0.137i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.107740096 + 0.07657914420i\)
\(L(\frac12)\) \(\approx\) \(1.107740096 + 0.07657914420i\)
\(L(1)\) \(\approx\) \(0.9608166809 + 0.04636307764i\)
\(L(1)\) \(\approx\) \(0.9608166809 + 0.04636307764i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (-0.997 + 0.0667i)T \)
3 \( 1 + (0.984 - 0.177i)T \)
5 \( 1 + (-0.770 + 0.637i)T \)
7 \( 1 + (0.756 + 0.654i)T \)
11 \( 1 + (-0.380 - 0.924i)T \)
13 \( 1 + (0.274 - 0.961i)T \)
17 \( 1 + (-0.122 - 0.992i)T \)
19 \( 1 + (0.695 + 0.718i)T \)
23 \( 1 + (0.519 + 0.854i)T \)
29 \( 1 + (0.100 + 0.994i)T \)
31 \( 1 + (0.975 + 0.220i)T \)
37 \( 1 + (-0.460 + 0.887i)T \)
41 \( 1 + (0.662 - 0.749i)T \)
43 \( 1 + (-0.420 + 0.907i)T \)
47 \( 1 + (-0.970 - 0.242i)T \)
53 \( 1 + (0.964 + 0.264i)T \)
59 \( 1 + (-0.929 - 0.369i)T \)
61 \( 1 + (0.231 - 0.972i)T \)
67 \( 1 + (-0.741 + 0.670i)T \)
71 \( 1 + (0.991 + 0.133i)T \)
73 \( 1 + (0.975 - 0.220i)T \)
79 \( 1 + (-0.420 - 0.907i)T \)
83 \( 1 + (0.0111 + 0.999i)T \)
89 \( 1 + (0.996 - 0.0890i)T \)
97 \( 1 + (-0.999 + 0.0222i)T \)
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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

−25.94401619726084048802798766508, −24.54387869855113477957956156917, −24.27005795818642104541458593663, −23.12693516563420130135476280812, −21.2309455806776831570374289200, −20.87039515396532420130265583349, −19.93709512802389351747880371166, −19.434290842470254658045654960462, −18.37925534931608252415161685752, −17.314568340324577102370662793442, −16.401306957611901633636692499296, −15.4530900057298032708700097023, −14.7963164541052388567761046720, −13.47806398262308107410855577428, −12.34517091660779870357280202202, −11.286222718999741681882719779158, −10.28987814542317114922115598656, −9.24470855437894221617482535450, −8.39074476329100893538319928598, −7.6938465410885418942330972652, −6.85190222460632624449725304633, −4.727292333838649542969604324821, −3.87372129869427215128077899068, −2.33722842191079350599338471879, −1.23211277367397713747402402729, 1.22220036403211420587085935774, 2.76688057033483490066235670441, 3.309699142000808578725226808273, 5.30983560160321787676696587842, 6.75197175211177617755131190501, 7.87688368187651228124069547212, 8.18640366995657428478913480068, 9.257097112013790939629688545786, 10.443281804332413155110587603255, 11.360174383400629917280707810006, 12.24687303525039014549525695350, 13.77120171268798498987033350722, 14.756639599724994054879306331745, 15.53930653370866584587154899856, 16.11866183617921184540924046863, 17.828023311412776796009095336733, 18.434527925809321095912055620652, 19.04377050475772549435536541932, 19.99705945875859406007418484502, 20.77157327607197564064309014055, 21.63979193858272709242942143875, 23.09071325815107556855894695927, 24.27293090712024958831435549875, 24.79909782368649665128628348875, 25.69339582843534074859067694212

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