Properties

Degree 1
Conductor 569
Sign $-0.316 - 0.948i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.448 + 0.894i)2-s + (0.387 + 0.921i)3-s + (−0.598 + 0.801i)4-s + (0.0663 + 0.997i)5-s + (−0.650 + 0.759i)6-s + (−0.903 + 0.428i)7-s + (−0.984 − 0.176i)8-s + (−0.699 + 0.714i)9-s + (−0.862 + 0.506i)10-s + (−0.0442 − 0.999i)11-s + (−0.970 − 0.240i)12-s + (0.839 + 0.544i)13-s + (−0.787 − 0.616i)14-s + (−0.894 + 0.448i)15-s + (−0.283 − 0.958i)16-s + (−0.814 − 0.580i)17-s + ⋯
L(s,χ)  = 1  + (0.448 + 0.894i)2-s + (0.387 + 0.921i)3-s + (−0.598 + 0.801i)4-s + (0.0663 + 0.997i)5-s + (−0.650 + 0.759i)6-s + (−0.903 + 0.428i)7-s + (−0.984 − 0.176i)8-s + (−0.699 + 0.714i)9-s + (−0.862 + 0.506i)10-s + (−0.0442 − 0.999i)11-s + (−0.970 − 0.240i)12-s + (0.839 + 0.544i)13-s + (−0.787 − 0.616i)14-s + (−0.894 + 0.448i)15-s + (−0.283 − 0.958i)16-s + (−0.814 − 0.580i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.316 - 0.948i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.316 - 0.948i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(569\)
\( \varepsilon \)  =  $-0.316 - 0.948i$
motivic weight  =  \(0\)
character  :  $\chi_{569} (28, \cdot )$
Sato-Tate  :  $\mu(284)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 569,\ (0:\ ),\ -0.316 - 0.948i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.6609011967 + 0.9172949799i$
$L(\frac12,\chi)$  $\approx$  $-0.6609011967 + 0.9172949799i$
$L(\chi,1)$  $\approx$  0.4476672591 + 1.001122085i
$L(1,\chi)$  $\approx$  0.4476672591 + 1.001122085i

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

−23.01484535265620239793355436744, −21.73121831754736277251672572577, −20.71630301285104289774403716951, −20.142583271571524232707882689308, −19.58435920870482364362775756330, −18.846502297176386589860908252606, −17.65723872069623800702662854086, −17.22628025938906259807988303749, −15.65344558676290631576818134233, −14.984776513155274740777765247393, −13.56584685805804071391998083279, −13.183567609923620379580193468224, −12.72345886609551074290088427057, −11.818107961087777471117924789712, −10.73876666994491609838637161915, −9.61135655845768086373408913806, −8.95757498928216243422104490784, −7.98282803988000727015187420380, −6.64069022029112275590706931485, −5.88411169722006500368804679593, −4.58179830818825123965803815156, −3.71491248542472442176466386752, −2.53819958543123268097002298381, −1.57654304024197827866487309228, −0.468775042716761350267133256943, 2.670426685673570160948107816581, 3.2842558062563799509586163490, 4.14235705469860634182628924931, 5.35651900844599977694739420358, 6.34671081501391932839033794764, 6.86466343717804749299145939632, 8.48938220208547159775545071224, 8.787043393599542413697311697524, 9.97784752319502175123175790630, 10.88574641539347120288756277468, 11.86674956010483928074283237883, 13.30265206682168633859925104725, 13.79377659995817890921996471629, 14.743175807555670815373918264092, 15.34387788082635905663660598268, 16.27221778361135855657198088862, 16.56241732563801087994161611289, 17.99184561453113989415738997962, 18.76975000711693220033297400923, 19.540411054018218163524306400548, 20.9743467480048986301182986535, 21.55199143341075047687432709084, 22.269178180162720706302217226007, 22.836213500062667959461329480742, 23.66748113117478262293898624154

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