Properties

Degree 1
Conductor 1021
Sign $-0.999 + 0.0351i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.631 − 0.775i)2-s + (−0.850 − 0.526i)3-s + (−0.201 − 0.979i)4-s + (−0.945 + 0.326i)5-s + (−0.945 + 0.326i)6-s + (−0.478 + 0.878i)7-s + (−0.886 − 0.462i)8-s + (0.445 + 0.895i)9-s + (−0.343 + 0.938i)10-s + (0.786 − 0.617i)11-s + (−0.343 + 0.938i)12-s + (0.932 − 0.361i)13-s + (0.378 + 0.925i)14-s + (0.975 + 0.219i)15-s + (−0.918 + 0.395i)16-s + (0.0184 + 0.999i)17-s + ⋯
L(s,χ)  = 1  + (0.631 − 0.775i)2-s + (−0.850 − 0.526i)3-s + (−0.201 − 0.979i)4-s + (−0.945 + 0.326i)5-s + (−0.945 + 0.326i)6-s + (−0.478 + 0.878i)7-s + (−0.886 − 0.462i)8-s + (0.445 + 0.895i)9-s + (−0.343 + 0.938i)10-s + (0.786 − 0.617i)11-s + (−0.343 + 0.938i)12-s + (0.932 − 0.361i)13-s + (0.378 + 0.925i)14-s + (0.975 + 0.219i)15-s + (−0.918 + 0.395i)16-s + (0.0184 + 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.999 + 0.0351i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.999 + 0.0351i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(1021\)
\( \varepsilon \)  =  $-0.999 + 0.0351i$
motivic weight  =  \(0\)
character  :  $\chi_{1021} (16, \cdot )$
Sato-Tate  :  $\mu(85)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 1021,\ (0:\ ),\ -0.999 + 0.0351i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.01346937759 - 0.7664377107i$
$L(\frac12,\chi)$  $\approx$  $0.01346937759 - 0.7664377107i$
$L(\chi,1)$  $\approx$  0.6614004702 - 0.5137094696i
$L(1,\chi)$  $\approx$  0.6614004702 - 0.5137094696i

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

−22.43579300350919116073224659646, −21.408573288089918113401824261987, −20.35553073162888213957452290432, −20.14913681438642644798994094470, −18.59222995367022699565394431505, −17.89772793653478419910933665849, −16.86376834623355078529350651134, −16.23207583960129996821810251387, −16.13057290875076602990580645764, −14.9950397711861442299014690141, −14.30172916434143397050007542275, −13.23562259381127278337115581923, −12.48442322461506031092537061288, −11.64078879106679966230964369089, −11.20586698987739653151875038065, −9.81211328038390008053621761477, −9.15306786274010652741299047491, −7.90323346712959292020335817088, −7.16062420434736672452015072170, −6.44448389087945163228209324871, −5.533359723623799151299397028309, −4.475408082735972317770920951583, −3.96810860857957892076478758402, −3.366002159923463260464523790864, −1.17609626989501702176222232892, 0.344534047888937092426867048437, 1.52527243197456638089857737942, 2.700548694331153748746929781223, 3.66477542693078986507590171536, 4.42366444086755162376143242969, 5.80791851073994290090061560612, 6.03566945258154398989688206800, 7.05144051136094518035224895002, 8.29714868838785446540177422725, 9.14775706417187450981563537334, 10.41253706128101212364170198557, 11.09394243291476435452353674659, 11.67687591577579417668358069614, 12.34043128351205607684855940169, 12.99237216538091151507047621548, 13.88045914681334601167021699077, 14.87947210525198000048364839062, 15.73363982839414071049476444984, 16.22407061891751313476779031896, 17.55655850105173813652647502706, 18.390242927624811212587353474064, 19.00163915023716016021524595841, 19.48379490908318166669792865358, 20.31417700991033677292574192810, 21.53917923102150926062523243380

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