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Properties

Label 1-1021-1021.135-r0-0-0
Degree $1$
Conductor $1021$
Sign $0.336 + 0.941i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.336 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.336 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $0.336 + 0.941i$
Analytic conductor: \(4.74150\)
Root analytic conductor: \(4.74150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (135, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (0:\ ),\ 0.336 + 0.941i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.473665021 + 1.037842109i\)
\(L(\frac12)\) \(\approx\) \(1.473665021 + 1.037842109i\)
\(L(1)\) \(\approx\) \(1.271956390 + 0.4026390714i\)
\(L(1)\) \(\approx\) \(1.271956390 + 0.4026390714i\)

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

−21.276157059121446401396045504413, −20.95801289002799055291187984397, −20.27934695867024849476537300819, −18.973571238091207474564568592790, −18.540841512066475265108617983507, −17.52269751077271704527524277109, −16.39025549887017832245237592533, −15.859506482678868763980849611672, −15.27661868514008145288770155395, −14.47687112049590987642617605791, −13.19544364679731765758253645772, −12.54488947980423572208603548020, −11.93208643218187586321835809907, −11.33484349914268557810447108988, −10.50717638340627051409323729984, −9.7714864383424704867916894224, −8.03745544071691176619835909770, −7.81004949811776222111449975628, −6.203125868290337356741075140246, −5.633447147203652506263379393550, −4.93860958845278199674022566077, −4.134324074004183158414167308828, −3.216610335693691931601599170675, −1.862337424967502244285131688838, −0.74909855871970613150559543823, 1.129169295907283469565291989513, 2.48980180389281386018979118087, 3.68713710749468874387650220835, 4.46604066370182987574979035081, 5.14799330395441437822893178017, 6.13007437134030604756133449714, 7.106481638610166374907021372964, 7.51075687017572650386252866612, 8.356375190561320816933673740709, 10.2005478797682783406617434902, 10.87212048864190466772066544141, 11.57588685481421371931390117206, 12.000034134960237080063294243528, 13.1325616938522854011512407537, 13.777445785711059488208568269273, 14.66068562462294020265631260442, 15.55948153795178185835983680033, 16.211936698405707426630606481894, 16.74712422966446250264674077052, 17.89744383777767167112546354212, 18.37589974145698024607327587012, 19.50543912537803604425220636410, 20.393586043636233891289215968840, 21.27951893807268822086661505044, 21.79882842413407559941768048543

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