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Properties

Label 1-1021-1021.44-r0-0-0
Degree $1$
Conductor $1021$
Sign $0.741 - 0.671i$
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  + (−0.572 − 0.819i)2-s + (−0.982 − 0.183i)3-s + (−0.343 + 0.938i)4-s + (0.995 + 0.0984i)5-s + (0.412 + 0.911i)6-s + (−0.989 + 0.147i)7-s + (0.966 − 0.255i)8-s + (0.932 + 0.361i)9-s + (−0.489 − 0.872i)10-s + (−0.320 − 0.947i)11-s + (0.510 − 0.859i)12-s + (0.602 + 0.798i)13-s + (0.687 + 0.726i)14-s + (−0.960 − 0.279i)15-s + (−0.763 − 0.645i)16-s + (0.592 + 0.805i)17-s + ⋯
L(s)  = 1  + (−0.572 − 0.819i)2-s + (−0.982 − 0.183i)3-s + (−0.343 + 0.938i)4-s + (0.995 + 0.0984i)5-s + (0.412 + 0.911i)6-s + (−0.989 + 0.147i)7-s + (0.966 − 0.255i)8-s + (0.932 + 0.361i)9-s + (−0.489 − 0.872i)10-s + (−0.320 − 0.947i)11-s + (0.510 − 0.859i)12-s + (0.602 + 0.798i)13-s + (0.687 + 0.726i)14-s + (−0.960 − 0.279i)15-s + (−0.763 − 0.645i)16-s + (0.592 + 0.805i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.741 - 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.741 - 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $0.741 - 0.671i$
Analytic conductor: \(4.74150\)
Root analytic conductor: \(4.74150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (0:\ ),\ 0.741 - 0.671i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7606531536 - 0.2932164161i\)
\(L(\frac12)\) \(\approx\) \(0.7606531536 - 0.2932164161i\)
\(L(1)\) \(\approx\) \(0.6368299950 - 0.2140917594i\)
\(L(1)\) \(\approx\) \(0.6368299950 - 0.2140917594i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1021 \( 1 \)
good2 \( 1 + (-0.572 - 0.819i)T \)
3 \( 1 + (-0.982 - 0.183i)T \)
5 \( 1 + (0.995 + 0.0984i)T \)
7 \( 1 + (-0.989 + 0.147i)T \)
11 \( 1 + (-0.320 - 0.947i)T \)
13 \( 1 + (0.602 + 0.798i)T \)
17 \( 1 + (0.592 + 0.805i)T \)
19 \( 1 + (0.153 - 0.988i)T \)
23 \( 1 + (0.941 + 0.338i)T \)
29 \( 1 + (-0.366 - 0.930i)T \)
31 \( 1 + (-0.892 + 0.451i)T \)
37 \( 1 + (0.0799 - 0.996i)T \)
41 \( 1 + (-0.153 + 0.988i)T \)
43 \( 1 + (0.249 + 0.968i)T \)
47 \( 1 + (0.285 - 0.958i)T \)
53 \( 1 + (-0.612 + 0.790i)T \)
59 \( 1 + (-0.923 - 0.384i)T \)
61 \( 1 + (-0.972 + 0.231i)T \)
67 \( 1 + (0.913 + 0.406i)T \)
71 \( 1 + (-0.966 + 0.255i)T \)
73 \( 1 + (0.687 - 0.726i)T \)
79 \( 1 + (0.975 + 0.219i)T \)
83 \( 1 + (0.332 + 0.943i)T \)
89 \( 1 + (0.969 - 0.243i)T \)
97 \( 1 + (0.998 + 0.0615i)T \)
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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

−22.07231279797415598306000849461, −20.69424978233085389779574390646, −20.358805161040395541378715407824, −18.75226468795649263222054853177, −18.49197458144566819716395276106, −17.64639157887166333216178049109, −16.94087336202279249248788479518, −16.40791121895863948874429901898, −15.66551074370047044574128625850, −14.85960765632475006430671239654, −13.8073906095182978115478845368, −12.94075947760875166626468409475, −12.37960402630800013019920772767, −10.83876154028655062789064938461, −10.3161676933946827063655919333, −9.657423446234670360371142296454, −9.04010990111142654711788147347, −7.59517472024462253805641135390, −6.91282431403149561162660400239, −6.07481082991042741276938524100, −5.47653163589599097706915075944, −4.7629350298468300556424686400, −3.34998844308779544744421960179, −1.783665915929493112798418775884, −0.73833023001526617718894732700, 0.79562348038407615322072616334, 1.76027248983399520264860175668, 2.85994475140333593720219258997, 3.79168022808423777556180255020, 5.06043937633577439748999419571, 6.01420808737728395779029638941, 6.64044632960066819823161320995, 7.68684223749430684528182253627, 9.04122420540234909544498776737, 9.432644239435067180401005266253, 10.49213495034216766603449957760, 10.95490863539508049140667370763, 11.77692693234577251889933151024, 12.91562040477508322535612588086, 13.12920379518510233754477222967, 13.98548023225260475028562524176, 15.55633358341289450185234321412, 16.56407798383089800911708992687, 16.779174944808638569276227582, 17.72357385151398332765146520485, 18.49737020133556668628083919501, 18.96446250321194920272459208436, 19.71679858635764099293158512050, 21.055839657045062052201558789196, 21.58156931861566738364842476412

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