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Properties

Label 1-283-283.152-r0-0-0
Degree $1$
Conductor $283$
Sign $0.358 + 0.933i$
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  + (−0.538 − 0.842i)2-s + (−0.892 + 0.451i)3-s + (−0.420 + 0.907i)4-s + (0.593 + 0.805i)5-s + (0.860 + 0.509i)6-s + (−0.296 + 0.955i)7-s + (0.991 − 0.133i)8-s + (0.593 − 0.805i)9-s + (0.359 − 0.933i)10-s + (−0.420 − 0.907i)11-s + (−0.0334 − 0.999i)12-s + (0.860 + 0.509i)13-s + (0.964 − 0.264i)14-s + (−0.892 − 0.451i)15-s + (−0.645 − 0.763i)16-s + (0.964 + 0.264i)17-s + ⋯
L(s)  = 1  + (−0.538 − 0.842i)2-s + (−0.892 + 0.451i)3-s + (−0.420 + 0.907i)4-s + (0.593 + 0.805i)5-s + (0.860 + 0.509i)6-s + (−0.296 + 0.955i)7-s + (0.991 − 0.133i)8-s + (0.593 − 0.805i)9-s + (0.359 − 0.933i)10-s + (−0.420 − 0.907i)11-s + (−0.0334 − 0.999i)12-s + (0.860 + 0.509i)13-s + (0.964 − 0.264i)14-s + (−0.892 − 0.451i)15-s + (−0.645 − 0.763i)16-s + (0.964 + 0.264i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.358 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.358 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $0.358 + 0.933i$
Analytic conductor: \(1.31424\)
Root analytic conductor: \(1.31424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (152, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (0:\ ),\ 0.358 + 0.933i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5425827096 + 0.3728131652i\)
\(L(\frac12)\) \(\approx\) \(0.5425827096 + 0.3728131652i\)
\(L(1)\) \(\approx\) \(0.6515732407 + 0.09077704734i\)
\(L(1)\) \(\approx\) \(0.6515732407 + 0.09077704734i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (-0.538 - 0.842i)T \)
3 \( 1 + (-0.892 + 0.451i)T \)
5 \( 1 + (0.593 + 0.805i)T \)
7 \( 1 + (-0.296 + 0.955i)T \)
11 \( 1 + (-0.420 - 0.907i)T \)
13 \( 1 + (0.860 + 0.509i)T \)
17 \( 1 + (0.964 + 0.264i)T \)
19 \( 1 + (0.860 + 0.509i)T \)
23 \( 1 + (-0.944 - 0.328i)T \)
29 \( 1 + (-0.997 + 0.0667i)T \)
31 \( 1 + (-0.979 + 0.199i)T \)
37 \( 1 + (0.784 + 0.619i)T \)
41 \( 1 + (0.991 + 0.133i)T \)
43 \( 1 + (0.231 + 0.972i)T \)
47 \( 1 + (0.860 - 0.509i)T \)
53 \( 1 + (-0.645 + 0.763i)T \)
59 \( 1 + (-0.824 - 0.565i)T \)
61 \( 1 + (0.359 + 0.933i)T \)
67 \( 1 + (-0.0334 + 0.999i)T \)
71 \( 1 + (-0.420 - 0.907i)T \)
73 \( 1 + (-0.979 - 0.199i)T \)
79 \( 1 + (0.231 - 0.972i)T \)
83 \( 1 + (-0.166 + 0.986i)T \)
89 \( 1 + (0.231 + 0.972i)T \)
97 \( 1 + (-0.944 - 0.328i)T \)
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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

−25.472764250020760501726927034441, −24.43545136065030580774885756475, −23.672795848896054840929078651569, −23.07571075531600737936254023042, −22.167297417066530635322962292660, −20.6245496641061937920361844549, −19.94125540412657717550235591436, −18.57440336353723285705111540402, −17.84618012163868959375089319079, −17.20853150368852349934444308076, −16.33187103025404112532644408429, −15.7816648560675447986778051721, −14.15696028152506338953694002482, −13.31885036131832491977879739801, −12.59213179005646076426023305176, −11.06750905152547678182965504006, −10.10328196916001365843811681383, −9.39249208018613752749482641017, −7.80626825612958416277523146815, −7.29158977262185105334216064289, −5.958094819498684815300572869739, −5.372054375107952740679714718164, −4.21522675665947905057639472821, −1.70698609222012741877917587909, −0.65723274800606975083166092703, 1.46413735492356128741361266259, 2.91900484094806089437252042821, 3.821314482558020680838299871613, 5.54902201787447029568575267668, 6.1692191548590934864304714515, 7.69888168782030002383567315008, 9.07600187125892355730043478226, 9.82553413515282832196438416253, 10.760795844740877408614565506312, 11.44953698642197522177560306822, 12.35940596068008140425165677484, 13.44328196240221915092971496671, 14.6186478720919303033770109921, 16.07869126098341174203607663049, 16.547052462851258690184904097237, 17.87752610143745128085239456069, 18.514166730176092160624817408599, 18.90492907692806649290133887856, 20.586346576028876237507528437687, 21.41044892907484519086467845304, 21.934970256466242660797720170266, 22.62311180610378449820999222756, 23.69597629480637511125881693716, 25.13519953518719812577100777030, 26.12184707223699891115537803770

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