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Properties

Label 1-1021-1021.121-r0-0-0
Degree $1$
Conductor $1021$
Sign $0.467 + 0.884i$
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  + (−0.945 + 0.326i)2-s + (−0.982 + 0.183i)3-s + (0.786 − 0.617i)4-s + (−0.862 − 0.505i)5-s + (0.869 − 0.494i)6-s + (−0.713 − 0.700i)7-s + (−0.542 + 0.840i)8-s + (0.932 − 0.361i)9-s + (0.980 + 0.195i)10-s + (−0.999 − 0.0123i)11-s + (−0.659 + 0.751i)12-s + (−0.602 + 0.798i)13-s + (0.903 + 0.429i)14-s + (0.941 + 0.338i)15-s + (0.237 − 0.971i)16-s + (0.949 + 0.314i)17-s + ⋯
L(s)  = 1  + (−0.945 + 0.326i)2-s + (−0.982 + 0.183i)3-s + (0.786 − 0.617i)4-s + (−0.862 − 0.505i)5-s + (0.869 − 0.494i)6-s + (−0.713 − 0.700i)7-s + (−0.542 + 0.840i)8-s + (0.932 − 0.361i)9-s + (0.980 + 0.195i)10-s + (−0.999 − 0.0123i)11-s + (−0.659 + 0.751i)12-s + (−0.602 + 0.798i)13-s + (0.903 + 0.429i)14-s + (0.941 + 0.338i)15-s + (0.237 − 0.971i)16-s + (0.949 + 0.314i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.467 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.467 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $0.467 + 0.884i$
Analytic conductor: \(4.74150\)
Root analytic conductor: \(4.74150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (0:\ ),\ 0.467 + 0.884i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1116455283 + 0.06728178827i\)
\(L(\frac12)\) \(\approx\) \(0.1116455283 + 0.06728178827i\)
\(L(1)\) \(\approx\) \(0.3144712591 + 0.01101294719i\)
\(L(1)\) \(\approx\) \(0.3144712591 + 0.01101294719i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1021 \( 1 \)
good2 \( 1 + (-0.945 + 0.326i)T \)
3 \( 1 + (-0.982 + 0.183i)T \)
5 \( 1 + (-0.862 - 0.505i)T \)
7 \( 1 + (-0.713 - 0.700i)T \)
11 \( 1 + (-0.999 - 0.0123i)T \)
13 \( 1 + (-0.602 + 0.798i)T \)
17 \( 1 + (0.949 + 0.314i)T \)
19 \( 1 + (-0.153 - 0.988i)T \)
23 \( 1 + (-0.960 - 0.279i)T \)
29 \( 1 + (-0.249 - 0.968i)T \)
31 \( 1 + (-0.987 + 0.159i)T \)
37 \( 1 + (-0.972 - 0.231i)T \)
41 \( 1 + (-0.153 - 0.988i)T \)
43 \( 1 + (-0.366 - 0.930i)T \)
47 \( 1 + (-0.823 + 0.567i)T \)
53 \( 1 + (-0.562 + 0.826i)T \)
59 \( 1 + (-0.521 + 0.853i)T \)
61 \( 1 + (-0.0799 - 0.996i)T \)
67 \( 1 + (-0.104 - 0.994i)T \)
71 \( 1 + (-0.542 + 0.840i)T \)
73 \( 1 + (0.903 - 0.429i)T \)
79 \( 1 + (-0.918 - 0.395i)T \)
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

−21.55373139008270385589205821521, −20.556700966799222552483576110869, −19.629074058116146390672650255341, −18.948209124144630391696636110674, −18.27247693712868014884276041144, −17.97320599896691850650272479241, −16.60085986471098675425388488104, −16.244789126420775079232035805816, −15.52449150533884134321719561976, −14.71881379130217329431203716682, −12.978716770198666073490438490977, −12.43293167059311188794329433119, −11.86755842248186722115446531941, −11.02302790067364685321071059184, −10.15970334515644064635969152832, −9.80625546433630584168841130682, −8.27489234156540934027381667819, −7.71378596703674823471581878903, −6.94704970748151936099713270378, −5.98078360990933785514832934123, −5.1542103302091433423771725159, −3.6049784857714452678943906921, −2.91690416787981491454171568626, −1.7171727803842462523509754585, −0.17521037052551145622475951277, 0.522755626499087355710665397718, 1.87632534238364074206978601547, 3.40743731476445621923532131644, 4.49563868955055906894958072797, 5.35696787155889978929815826590, 6.31986468808700604132999442850, 7.27972123409011766365715217458, 7.665502430172473534430399066211, 8.89777010018767240943555415551, 9.77074041425656985587996719373, 10.461413923378157380639155086405, 11.16298695142520115317503802443, 12.08231534969366608408962757407, 12.654213581620225385012418249370, 13.871896004350989384941719748148, 15.15243707236693952600833591335, 15.7978122355015911315122229008, 16.376001293433434503340081239158, 16.936259389220623152728053846484, 17.61037358397846738529189230950, 18.75196107734721487223355926658, 19.12347175412285831181398657697, 20.06125787066574204503585929999, 20.774133022694126834693484543618, 21.68293793896191935870231854734

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