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Properties

Label 1-1021-1021.126-r0-0-0
Degree $1$
Conductor $1021$
Sign $0.588 + 0.808i$
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  + (−0.542 + 0.840i)2-s + (−0.850 + 0.526i)3-s + (−0.412 − 0.911i)4-s + (0.0184 − 0.999i)5-s + (0.0184 − 0.999i)6-s + (0.687 − 0.726i)7-s + (0.989 + 0.147i)8-s + (0.445 − 0.895i)9-s + (0.830 + 0.557i)10-s + (−0.999 − 0.0369i)11-s + (0.830 + 0.557i)12-s + (0.932 + 0.361i)13-s + (0.237 + 0.971i)14-s + (0.510 + 0.859i)15-s + (−0.659 + 0.751i)16-s + (0.572 + 0.819i)17-s + ⋯
L(s)  = 1  + (−0.542 + 0.840i)2-s + (−0.850 + 0.526i)3-s + (−0.412 − 0.911i)4-s + (0.0184 − 0.999i)5-s + (0.0184 − 0.999i)6-s + (0.687 − 0.726i)7-s + (0.989 + 0.147i)8-s + (0.445 − 0.895i)9-s + (0.830 + 0.557i)10-s + (−0.999 − 0.0369i)11-s + (0.830 + 0.557i)12-s + (0.932 + 0.361i)13-s + (0.237 + 0.971i)14-s + (0.510 + 0.859i)15-s + (−0.659 + 0.751i)16-s + (0.572 + 0.819i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.588 + 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.588 + 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $0.588 + 0.808i$
Analytic conductor: \(4.74150\)
Root analytic conductor: \(4.74150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (126, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (0:\ ),\ 0.588 + 0.808i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7312945404 + 0.3724479828i\)
\(L(\frac12)\) \(\approx\) \(0.7312945404 + 0.3724479828i\)
\(L(1)\) \(\approx\) \(0.6473109235 + 0.2040552155i\)
\(L(1)\) \(\approx\) \(0.6473109235 + 0.2040552155i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1021 \( 1 \)
good2 \( 1 + (-0.542 + 0.840i)T \)
3 \( 1 + (-0.850 + 0.526i)T \)
5 \( 1 + (0.0184 - 0.999i)T \)
7 \( 1 + (0.687 - 0.726i)T \)
11 \( 1 + (-0.999 - 0.0369i)T \)
13 \( 1 + (0.932 + 0.361i)T \)
17 \( 1 + (0.572 + 0.819i)T \)
19 \( 1 + (0.445 + 0.895i)T \)
23 \( 1 + (-0.659 - 0.751i)T \)
29 \( 1 + (0.687 + 0.726i)T \)
31 \( 1 + (-0.886 + 0.462i)T \)
37 \( 1 + (-0.763 - 0.645i)T \)
41 \( 1 + (0.445 + 0.895i)T \)
43 \( 1 + (0.903 + 0.429i)T \)
47 \( 1 + (0.237 + 0.971i)T \)
53 \( 1 + (0.975 + 0.219i)T \)
59 \( 1 + (0.997 + 0.0738i)T \)
61 \( 1 + (0.237 + 0.971i)T \)
67 \( 1 + (0.309 + 0.951i)T \)
71 \( 1 + (0.989 + 0.147i)T \)
73 \( 1 + (0.237 - 0.971i)T \)
79 \( 1 + (-0.343 - 0.938i)T \)
83 \( 1 + (-0.850 + 0.526i)T \)
89 \( 1 + (0.739 + 0.673i)T \)
97 \( 1 + (-0.982 + 0.183i)T \)
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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

−21.4615390930817693435457491820, −20.85353908494932445478214334116, −19.775530727583694531913099713492, −18.62188906625086935302217127256, −18.56367448194621169191874678129, −17.838981751122897361422513769820, −17.30208006869539077405550099589, −15.92035201861164071703566755846, −15.54578631899973773334644647659, −13.98189096162734231642467696158, −13.48974866596344317864777660456, −12.46350705300690181060805082064, −11.63655063611925269203048748709, −11.20658828820347430481108604404, −10.48317864641207418166249613687, −9.690114438429438813281617227198, −8.43068361823741578846545808496, −7.684076741792382457023822116999, −7.02786194658631817932726442380, −5.72080821872628596024252808715, −5.10974426955303922419439253723, −3.73578659080032385750590567157, −2.61056507718363286373870487060, −2.00818141810951246101250683769, −0.68047375133938586938135525692, 0.876489759166555451565161795065, 1.588490753848174076958377769947, 3.87725677368856235574262470337, 4.4997609536079115660892450133, 5.462230630784773370048480422163, 5.87419554086962573339455066556, 7.076631083801960999568653457215, 8.04856823937856356903719639273, 8.60815198029280252248561119861, 9.68171716181106593707178174200, 10.488461255751956603117111766779, 10.94937129790541428446900551854, 12.18235447314932701102752165874, 13.01068950663782596536753698605, 14.033069871935626885401726451022, 14.75921010565460555131620285369, 15.969038620098620883052926281617, 16.241815902920758360349672241934, 16.80777891265049023581716633705, 17.866609531210516769636374975926, 18.04494304564736763676341787335, 19.2198999047183468784124580128, 20.37182653462225282908719281674, 20.85781995777235071674905605358, 21.61704463748516832169948839636

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