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Properties

Label 1-1021-1021.7-r1-0-0
Degree $1$
Conductor $1021$
Sign $-0.805 - 0.593i$
Analytic cond. $109.721$
Root an. cond. $109.721$
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  + (−0.494 + 0.869i)2-s + (−0.445 + 0.895i)3-s + (−0.510 − 0.859i)4-s + (0.830 − 0.557i)5-s + (−0.557 − 0.830i)6-s + (0.775 − 0.631i)7-s + (0.999 − 0.0184i)8-s + (−0.602 − 0.798i)9-s + (0.0738 + 0.997i)10-s + (0.378 − 0.925i)11-s + (0.997 − 0.0738i)12-s + (−0.673 + 0.739i)13-s + (0.165 + 0.986i)14-s + (0.128 + 0.991i)15-s + (−0.478 + 0.878i)16-s + (−0.786 − 0.617i)17-s + ⋯
L(s)  = 1  + (−0.494 + 0.869i)2-s + (−0.445 + 0.895i)3-s + (−0.510 − 0.859i)4-s + (0.830 − 0.557i)5-s + (−0.557 − 0.830i)6-s + (0.775 − 0.631i)7-s + (0.999 − 0.0184i)8-s + (−0.602 − 0.798i)9-s + (0.0738 + 0.997i)10-s + (0.378 − 0.925i)11-s + (0.997 − 0.0738i)12-s + (−0.673 + 0.739i)13-s + (0.165 + 0.986i)14-s + (0.128 + 0.991i)15-s + (−0.478 + 0.878i)16-s + (−0.786 − 0.617i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.805 - 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.805 - 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $-0.805 - 0.593i$
Analytic conductor: \(109.721\)
Root analytic conductor: \(109.721\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (1:\ ),\ -0.805 - 0.593i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04013986505 - 0.1221423403i\)
\(L(\frac12)\) \(\approx\) \(0.04013986505 - 0.1221423403i\)
\(L(1)\) \(\approx\) \(0.6706538907 + 0.2038178583i\)
\(L(1)\) \(\approx\) \(0.6706538907 + 0.2038178583i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1021 \( 1 \)
good2 \( 1 + (-0.494 + 0.869i)T \)
3 \( 1 + (-0.445 + 0.895i)T \)
5 \( 1 + (0.830 - 0.557i)T \)
7 \( 1 + (0.775 - 0.631i)T \)
11 \( 1 + (0.378 - 0.925i)T \)
13 \( 1 + (-0.673 + 0.739i)T \)
17 \( 1 + (-0.786 - 0.617i)T \)
19 \( 1 + (-0.798 - 0.602i)T \)
23 \( 1 + (-0.478 - 0.878i)T \)
29 \( 1 + (-0.631 + 0.775i)T \)
31 \( 1 + (0.326 + 0.945i)T \)
37 \( 1 + (-0.462 - 0.886i)T \)
41 \( 1 + (0.602 - 0.798i)T \)
43 \( 1 + (-0.998 + 0.0554i)T \)
47 \( 1 + (0.165 + 0.986i)T \)
53 \( 1 + (0.726 + 0.687i)T \)
59 \( 1 + (0.700 - 0.713i)T \)
61 \( 1 + (-0.165 - 0.986i)T \)
67 \( 1 + (-0.809 - 0.587i)T \)
71 \( 1 + (0.0184 + 0.999i)T \)
73 \( 1 + (0.165 - 0.986i)T \)
79 \( 1 + (-0.237 + 0.971i)T \)
83 \( 1 + (-0.445 + 0.895i)T \)
89 \( 1 + (0.0922 + 0.995i)T \)
97 \( 1 + (0.361 + 0.932i)T \)
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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

−21.73645806132241020244557641042, −20.94927696639659602035485537002, −20.00559991498643030373548035027, −19.29283626331024843652389547944, −18.45743904107547328185530827045, −17.90605664257282479695418844720, −17.34038632645118676550167918810, −16.91995015675500361515924701947, −15.1699516997655773594858937751, −14.571233822451964122805781337325, −13.36815346646790249463175918195, −12.99583419486783546895532441480, −11.90981641723403786065133348012, −11.55587662696426567808290679068, −10.45625854930254649984017492333, −9.92263666039055829819919388963, −8.80128965361055075976497358590, −7.95820389237561419267417265090, −7.21391960527557606064829770903, −6.15516352963468736024591775856, −5.28599120345970206839794038463, −4.22178369684834661887161116755, −2.72201030414131915751561418133, −1.99155833849148483611263202536, −1.55900131268071661527706254931, 0.03649633559021987703857462153, 0.983648092886100567339370660956, 2.216938128671122210266481957095, 4.033386895962833836066269104916, 4.74552027547116037696499194303, 5.3236363062588565185833894317, 6.357476799077497525393906228889, 6.99151473187072585016465457107, 8.42140491127324516264108874653, 8.95213136489668056816186632570, 9.60715398767399355132369509660, 10.66608056552312487569953714097, 11.02831564066579060737626792654, 12.2900611727934282425124792680, 13.589268780072939327318948898360, 14.206108449706764777001192682232, 14.74110217609331775622851259251, 15.945329523395389281464029317173, 16.45793521985609380096317558032, 17.15851894006112041151521928915, 17.563915576976262679682025369053, 18.40541142204317630179021041186, 19.62332030870289059350880551307, 20.30093494991938625380301212289, 21.25731745638160355281974914770

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