Properties

Degree 1
Conductor 1021
Sign $0.595 + 0.803i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.830 + 0.557i)2-s + (0.932 − 0.361i)3-s + (0.378 + 0.925i)4-s + (0.975 + 0.219i)5-s + (0.975 + 0.219i)6-s + (−0.945 + 0.326i)7-s + (−0.201 + 0.979i)8-s + (0.739 − 0.673i)9-s + (0.687 + 0.726i)10-s + (0.903 + 0.429i)11-s + (0.687 + 0.726i)12-s + (−0.273 − 0.961i)13-s + (−0.966 − 0.255i)14-s + (0.989 − 0.147i)15-s + (−0.713 + 0.700i)16-s + (0.510 − 0.859i)17-s + ⋯
L(s,χ)  = 1  + (0.830 + 0.557i)2-s + (0.932 − 0.361i)3-s + (0.378 + 0.925i)4-s + (0.975 + 0.219i)5-s + (0.975 + 0.219i)6-s + (−0.945 + 0.326i)7-s + (−0.201 + 0.979i)8-s + (0.739 − 0.673i)9-s + (0.687 + 0.726i)10-s + (0.903 + 0.429i)11-s + (0.687 + 0.726i)12-s + (−0.273 − 0.961i)13-s + (−0.966 − 0.255i)14-s + (0.989 − 0.147i)15-s + (−0.713 + 0.700i)16-s + (0.510 − 0.859i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.595 + 0.803i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.595 + 0.803i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(1021\)
\( \varepsilon \)  =  $0.595 + 0.803i$
motivic weight  =  \(0\)
character  :  $\chi_{1021} (196, \cdot )$
Sato-Tate  :  $\mu(85)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 1021,\ (0:\ ),\ 0.595 + 0.803i)$
$L(\chi,\frac{1}{2})$  $\approx$  $3.519407939 + 1.771437481i$
$L(\frac12,\chi)$  $\approx$  $3.519407939 + 1.771437481i$
$L(\chi,1)$  $\approx$  2.363763698 + 0.7872703473i
$L(1,\chi)$  $\approx$  2.363763698 + 0.7872703473i

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

−21.442139387856077321243271693221, −20.887150441117555252971641886596, −19.94594354407955431055759197232, −19.395833661777443850506984930913, −18.89059835568523851486008083120, −17.55069296763561400791329983883, −16.45777327812345404551974230739, −15.99129707946213672695934668814, −14.81620763979857054108110764357, −14.17404633381610982512319637730, −13.65116957966491520994604616726, −13.00197987549476026699072587270, −12.144644228157717843936811253765, −11.03779834259622878327095348485, −10.11458304679457724684426277286, −9.40404829737408224526584562042, −9.12669687510857596173782591827, −7.512125370703512390724414775791, −6.50786725795566542003472067301, −5.82987162962336551907751120820, −4.67349101285544785313165475767, −3.80395316933038030575944558919, −3.15230112975499669055353983642, −2.089358371274890182283995885664, −1.310879304402346177036110390433, 1.50573248856269652247913057089, 2.7075745960455330663091717912, 3.094307555209354131967947208149, 4.167572519888369228005272920290, 5.42435844129793773803763271992, 6.191746884289035473813411801744, 6.91515312634912109717315013671, 7.687752105543391395842877418562, 8.72046709024898828183093089401, 9.60712670672556119574780166999, 10.15653115295942718434772257049, 11.848851147602929053668535933869, 12.46952262173521544871939045868, 13.16007935133782500968875268858, 13.93859685540023186843626558364, 14.44202556045501535004457651715, 15.21312915931747818521399361032, 16.04484597283084173473809762698, 16.901751061569791909326534803793, 17.83059961085163177960788639183, 18.49551194967421941823264083775, 19.52392367372868996945808065672, 20.45994883676716371546699588610, 20.80378626430153543639428266714, 21.971389964919533354370064555117

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