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Properties

Label 1-1021-1021.224-r0-0-0
Degree $1$
Conductor $1021$
Sign $0.595 - 0.803i$
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  + (0.830 − 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $0.595 - 0.803i$
Analytic conductor: \(4.74150\)
Root analytic conductor: \(4.74150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (224, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (0:\ ),\ 0.595 - 0.803i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.519407939 - 1.771437481i\)
\(L(\frac12)\) \(\approx\) \(3.519407939 - 1.771437481i\)
\(L(1)\) \(\approx\) \(2.363763698 - 0.7872703473i\)
\(L(1)\) \(\approx\) \(2.363763698 - 0.7872703473i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1021 \( 1 \)
good2 \( 1 + (0.830 - 0.557i)T \)
3 \( 1 + (0.932 + 0.361i)T \)
5 \( 1 + (0.975 - 0.219i)T \)
7 \( 1 + (-0.945 - 0.326i)T \)
11 \( 1 + (0.903 - 0.429i)T \)
13 \( 1 + (-0.273 + 0.961i)T \)
17 \( 1 + (0.510 + 0.859i)T \)
19 \( 1 + (0.739 - 0.673i)T \)
23 \( 1 + (-0.713 + 0.700i)T \)
29 \( 1 + (-0.945 + 0.326i)T \)
31 \( 1 + (0.869 - 0.494i)T \)
37 \( 1 + (-0.542 - 0.840i)T \)
41 \( 1 + (0.739 - 0.673i)T \)
43 \( 1 + (0.572 + 0.819i)T \)
47 \( 1 + (-0.966 + 0.255i)T \)
53 \( 1 + (-0.886 - 0.462i)T \)
59 \( 1 + (0.631 - 0.775i)T \)
61 \( 1 + (-0.966 + 0.255i)T \)
67 \( 1 + (-0.809 - 0.587i)T \)
71 \( 1 + (-0.201 - 0.979i)T \)
73 \( 1 + (-0.966 - 0.255i)T \)
79 \( 1 + (-0.478 - 0.878i)T \)
83 \( 1 + (0.932 + 0.361i)T \)
89 \( 1 + (-0.850 - 0.526i)T \)
97 \( 1 + (-0.602 + 0.798i)T \)
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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

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

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