Properties

Degree 1
Conductor 283
Sign $0.0710 - 0.997i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(283\)
\( \varepsilon \)  =  $0.0710 - 0.997i$
motivic weight  =  \(0\)
character  :  $\chi_{283} (42, \cdot )$
Sato-Tate  :  $\mu(47)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 283,\ (0:\ ),\ 0.0710 - 0.997i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.434282323 - 1.335693235i$
$L(\frac12,\chi)$  $\approx$  $1.434282323 - 1.335693235i$
$L(\chi,1)$  $\approx$  1.446793817 - 0.7343037497i
$L(1,\chi)$  $\approx$  1.446793817 - 0.7343037497i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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