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Properties

Label 1-283-283.155-r0-0-0
Degree $1$
Conductor $283$
Sign $0.0710 + 0.997i$
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  + (0.964 + 0.264i)2-s + (−0.944 + 0.328i)3-s + (0.860 + 0.509i)4-s + (0.784 + 0.619i)5-s + (−0.997 + 0.0667i)6-s + (0.231 + 0.972i)7-s + (0.695 + 0.718i)8-s + (0.784 − 0.619i)9-s + (0.593 + 0.805i)10-s + (0.860 − 0.509i)11-s + (−0.979 − 0.199i)12-s + (−0.997 + 0.0667i)13-s + (−0.0334 + 0.999i)14-s + (−0.944 − 0.328i)15-s + (0.480 + 0.876i)16-s + (−0.0334 − 0.999i)17-s + ⋯
L(s)  = 1  + (0.964 + 0.264i)2-s + (−0.944 + 0.328i)3-s + (0.860 + 0.509i)4-s + (0.784 + 0.619i)5-s + (−0.997 + 0.0667i)6-s + (0.231 + 0.972i)7-s + (0.695 + 0.718i)8-s + (0.784 − 0.619i)9-s + (0.593 + 0.805i)10-s + (0.860 − 0.509i)11-s + (−0.979 − 0.199i)12-s + (−0.997 + 0.0667i)13-s + (−0.0334 + 0.999i)14-s + (−0.944 − 0.328i)15-s + (0.480 + 0.876i)16-s + (−0.0334 − 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0710 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0710 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $0.0710 + 0.997i$
Analytic conductor: \(1.31424\)
Root analytic conductor: \(1.31424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (0:\ ),\ 0.0710 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.434282323 + 1.335693235i\)
\(L(\frac12)\) \(\approx\) \(1.434282323 + 1.335693235i\)
\(L(1)\) \(\approx\) \(1.446793817 + 0.7343037497i\)
\(L(1)\) \(\approx\) \(1.446793817 + 0.7343037497i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (0.964 + 0.264i)T \)
3 \( 1 + (-0.944 + 0.328i)T \)
5 \( 1 + (0.784 + 0.619i)T \)
7 \( 1 + (0.231 + 0.972i)T \)
11 \( 1 + (0.860 - 0.509i)T \)
13 \( 1 + (-0.997 + 0.0667i)T \)
17 \( 1 + (-0.0334 - 0.999i)T \)
19 \( 1 + (-0.997 + 0.0667i)T \)
23 \( 1 + (-0.420 - 0.907i)T \)
29 \( 1 + (0.920 + 0.390i)T \)
31 \( 1 + (0.359 + 0.933i)T \)
37 \( 1 + (-0.645 + 0.763i)T \)
41 \( 1 + (0.695 - 0.718i)T \)
43 \( 1 + (-0.166 - 0.986i)T \)
47 \( 1 + (-0.997 - 0.0667i)T \)
53 \( 1 + (0.480 - 0.876i)T \)
59 \( 1 + (-0.892 + 0.451i)T \)
61 \( 1 + (0.593 - 0.805i)T \)
67 \( 1 + (-0.979 + 0.199i)T \)
71 \( 1 + (0.860 - 0.509i)T \)
73 \( 1 + (0.359 - 0.933i)T \)
79 \( 1 + (-0.166 + 0.986i)T \)
83 \( 1 + (-0.538 + 0.842i)T \)
89 \( 1 + (-0.166 - 0.986i)T \)
97 \( 1 + (-0.420 - 0.907i)T \)
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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

−24.8599032588337045525107645904, −24.33735597669532489847334314176, −23.466589636855199535462965822079, −22.74462452184886483257196127846, −21.70122825267520080827023299233, −21.2301929237489031767504065165, −19.909162093959542334142568857190, −19.38463066569916361275667834754, −17.58708209663241563416321524098, −17.1789729548829442033819981658, −16.35306492617292147700205707866, −14.983157805002076295130614238720, −14.01133745208331142283370691591, −13.07877695336494579153541216801, −12.44075388548499172325546970174, −11.49646794417289372239887430852, −10.40557617767444541178850901976, −9.737010728425384680004471675014, −7.770197735075279659696350050391, −6.6560208082745613032430880734, −5.93526537428463708430644493291, −4.7232541889536640755213927542, −4.16092849909961349484731125938, −2.1000757749097180939945507926, −1.208572769421558928792869297474, 1.95390027606834993040440315668, 3.095393499147116037913310602117, 4.60037807329367247666095455070, 5.387371152169452394450569087460, 6.37135787194674850376036634038, 6.915239267808626521564088480646, 8.69189491654737715036034793149, 9.99441624235111223999542514850, 10.98445254051365665307331398580, 11.939873750925812293363484009830, 12.50925114279909949360610864940, 13.95212509469760924601780524197, 14.64633753096656593345349687049, 15.54418647118146517637824278835, 16.578583213670469062544280546061, 17.35325829287844620918558818868, 18.25302905542791081584680169275, 19.43175435708009925088485615603, 20.991035597048274919420536734508, 21.54988200181319641651206800927, 22.30705503757538500168068330957, 22.680198708205490268061294723147, 24.0235456689396497376352545858, 24.733572570415781472670261961212, 25.46537010447288002558036477533

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