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Properties

Label 1-1021-1021.147-r0-0-0
Degree $1$
Conductor $1021$
Sign $0.451 - 0.892i$
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  + (0.378 − 0.925i)2-s + (0.739 + 0.673i)3-s + (−0.713 − 0.700i)4-s + (0.903 − 0.429i)5-s + (0.903 − 0.429i)6-s + (0.786 + 0.617i)7-s + (−0.918 + 0.395i)8-s + (0.0922 + 0.995i)9-s + (−0.0554 − 0.998i)10-s + (0.631 − 0.775i)11-s + (−0.0554 − 0.998i)12-s + (−0.850 − 0.526i)13-s + (0.869 − 0.494i)14-s + (0.956 + 0.291i)15-s + (0.0184 + 0.999i)16-s + (−0.478 + 0.878i)17-s + ⋯
L(s)  = 1  + (0.378 − 0.925i)2-s + (0.739 + 0.673i)3-s + (−0.713 − 0.700i)4-s + (0.903 − 0.429i)5-s + (0.903 − 0.429i)6-s + (0.786 + 0.617i)7-s + (−0.918 + 0.395i)8-s + (0.0922 + 0.995i)9-s + (−0.0554 − 0.998i)10-s + (0.631 − 0.775i)11-s + (−0.0554 − 0.998i)12-s + (−0.850 − 0.526i)13-s + (0.869 − 0.494i)14-s + (0.956 + 0.291i)15-s + (0.0184 + 0.999i)16-s + (−0.478 + 0.878i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.451 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.451 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $0.451 - 0.892i$
Analytic conductor: \(4.74150\)
Root analytic conductor: \(4.74150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (0:\ ),\ 0.451 - 0.892i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.385381848 - 1.466681831i\)
\(L(\frac12)\) \(\approx\) \(2.385381848 - 1.466681831i\)
\(L(1)\) \(\approx\) \(1.716717812 - 0.6827326098i\)
\(L(1)\) \(\approx\) \(1.716717812 - 0.6827326098i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1021 \( 1 \)
good2 \( 1 + (0.378 - 0.925i)T \)
3 \( 1 + (0.739 + 0.673i)T \)
5 \( 1 + (0.903 - 0.429i)T \)
7 \( 1 + (0.786 + 0.617i)T \)
11 \( 1 + (0.631 - 0.775i)T \)
13 \( 1 + (-0.850 - 0.526i)T \)
17 \( 1 + (-0.478 + 0.878i)T \)
19 \( 1 + (0.0922 - 0.995i)T \)
23 \( 1 + (0.0184 - 0.999i)T \)
29 \( 1 + (0.786 - 0.617i)T \)
31 \( 1 + (0.510 - 0.859i)T \)
37 \( 1 + (-0.412 + 0.911i)T \)
41 \( 1 + (0.0922 - 0.995i)T \)
43 \( 1 + (-0.343 + 0.938i)T \)
47 \( 1 + (0.869 - 0.494i)T \)
53 \( 1 + (0.572 + 0.819i)T \)
59 \( 1 + (-0.201 - 0.979i)T \)
61 \( 1 + (0.869 - 0.494i)T \)
67 \( 1 + (0.309 + 0.951i)T \)
71 \( 1 + (-0.918 + 0.395i)T \)
73 \( 1 + (0.869 + 0.494i)T \)
79 \( 1 + (-0.542 + 0.840i)T \)
83 \( 1 + (0.739 + 0.673i)T \)
89 \( 1 + (0.445 + 0.895i)T \)
97 \( 1 + (-0.273 - 0.961i)T \)
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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

−21.71602040081729226149359309888, −21.09196493918892679152330151169, −20.26462999278801725595004203889, −19.3453037122665147142686162985, −18.28770424996146410464698107553, −17.6965413342878165849794513289, −17.309838750733854423350907431530, −16.26848983295726335667205348053, −15.03361238273925806798442649969, −14.509884166109579514131194867107, −13.96623268461954085361891373815, −13.482857711773565474603178302232, −12.36289363139473577739179218064, −11.754843916936991567024977369459, −10.215687471224799108876910300501, −9.41554282007835602811208247250, −8.72423239300246808106732861479, −7.55001965524914666718272761764, −7.13099838821760911273145006048, −6.46667779317564983713280646892, −5.2971299452714483600358954260, −4.42190484149773356801682189101, −3.39639953620630520714107728062, −2.27866660159400110466708152507, −1.35492025149538496615487722533, 1.087811945626046009768229697917, 2.34916665957597939613236159468, 2.607300400250360328020138570107, 3.997149150521785870936303270257, 4.7754087586148839665122647929, 5.420648411290513922942681179428, 6.41156736643234151046747284240, 8.25385594151098212372200022787, 8.68362606982945598327117226448, 9.456041512451358996136806874883, 10.23138157593146174991167492515, 10.96568716973043376633161448765, 11.89087050960081355823179051493, 12.82121334585348647883985987421, 13.61329769216449079636385731894, 14.23912238793005535299839123901, 14.93630970533845694795777129602, 15.62975991244784576417523830382, 17.06312525000810383768942449541, 17.48299124625600704073219835058, 18.61776988215194246775153441872, 19.348723005036018342211291859367, 20.139741790341510944020228773870, 20.7325182957048234392424049068, 21.528295995642833277420286577386

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