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Properties

Label 1-1021-1021.16-r0-0-0
Degree $1$
Conductor $1021$
Sign $-0.999 + 0.0351i$
Analytic cond. $4.74150$
Root an. cond. $4.74150$
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  + (0.631 − 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $-0.999 + 0.0351i$
Analytic conductor: \(4.74150\)
Root analytic conductor: \(4.74150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (0:\ ),\ -0.999 + 0.0351i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01346937759 - 0.7664377107i\)
\(L(\frac12)\) \(\approx\) \(0.01346937759 - 0.7664377107i\)
\(L(1)\) \(\approx\) \(0.6614004702 - 0.5137094696i\)
\(L(1)\) \(\approx\) \(0.6614004702 - 0.5137094696i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1021 \( 1 \)
good2 \( 1 + (0.631 - 0.775i)T \)
3 \( 1 + (-0.850 - 0.526i)T \)
5 \( 1 + (-0.945 + 0.326i)T \)
7 \( 1 + (-0.478 + 0.878i)T \)
11 \( 1 + (0.786 - 0.617i)T \)
13 \( 1 + (0.932 - 0.361i)T \)
17 \( 1 + (0.0184 + 0.999i)T \)
19 \( 1 + (0.445 - 0.895i)T \)
23 \( 1 + (-0.918 - 0.395i)T \)
29 \( 1 + (-0.478 - 0.878i)T \)
31 \( 1 + (-0.713 + 0.700i)T \)
37 \( 1 + (0.997 - 0.0738i)T \)
41 \( 1 + (0.445 - 0.895i)T \)
43 \( 1 + (-0.128 - 0.991i)T \)
47 \( 1 + (0.378 + 0.925i)T \)
53 \( 1 + (-0.659 + 0.751i)T \)
59 \( 1 + (0.237 - 0.971i)T \)
61 \( 1 + (0.378 + 0.925i)T \)
67 \( 1 + (-0.809 + 0.587i)T \)
71 \( 1 + (-0.886 - 0.462i)T \)
73 \( 1 + (0.378 - 0.925i)T \)
79 \( 1 + (-0.999 - 0.0369i)T \)
83 \( 1 + (-0.850 - 0.526i)T \)
89 \( 1 + (0.739 - 0.673i)T \)
97 \( 1 + (-0.982 - 0.183i)T \)
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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

−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