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Properties

Label 1-1021-1021.465-r1-0-0
Degree $1$
Conductor $1021$
Sign $-0.954 - 0.297i$
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  + (0.798 + 0.602i)2-s + (0.982 + 0.183i)3-s + (0.273 + 0.961i)4-s + (0.739 − 0.673i)5-s + (0.673 + 0.739i)6-s + (−0.895 + 0.445i)7-s + (−0.361 + 0.932i)8-s + (0.932 + 0.361i)9-s + (0.995 − 0.0922i)10-s + (0.0922 − 0.995i)11-s + (0.0922 + 0.995i)12-s + (−0.798 + 0.602i)13-s + (−0.982 − 0.183i)14-s + (0.850 − 0.526i)15-s + (−0.850 + 0.526i)16-s + (−0.739 + 0.673i)17-s + ⋯
L(s)  = 1  + (0.798 + 0.602i)2-s + (0.982 + 0.183i)3-s + (0.273 + 0.961i)4-s + (0.739 − 0.673i)5-s + (0.673 + 0.739i)6-s + (−0.895 + 0.445i)7-s + (−0.361 + 0.932i)8-s + (0.932 + 0.361i)9-s + (0.995 − 0.0922i)10-s + (0.0922 − 0.995i)11-s + (0.0922 + 0.995i)12-s + (−0.798 + 0.602i)13-s + (−0.982 − 0.183i)14-s + (0.850 − 0.526i)15-s + (−0.850 + 0.526i)16-s + (−0.739 + 0.673i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.954 - 0.297i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.954 - 0.297i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $-0.954 - 0.297i$
Analytic conductor: \(109.721\)
Root analytic conductor: \(109.721\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (465, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (1:\ ),\ -0.954 - 0.297i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3137946939 + 2.059184210i\)
\(L(\frac12)\) \(\approx\) \(-0.3137946939 + 2.059184210i\)
\(L(1)\) \(\approx\) \(1.567188861 + 0.9212449086i\)
\(L(1)\) \(\approx\) \(1.567188861 + 0.9212449086i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1021 \( 1 \)
good2 \( 1 + (0.798 + 0.602i)T \)
3 \( 1 + (0.982 + 0.183i)T \)
5 \( 1 + (0.739 - 0.673i)T \)
7 \( 1 + (-0.895 + 0.445i)T \)
11 \( 1 + (0.0922 - 0.995i)T \)
13 \( 1 + (-0.798 + 0.602i)T \)
17 \( 1 + (-0.739 + 0.673i)T \)
19 \( 1 + (-0.361 - 0.932i)T \)
23 \( 1 + (-0.850 - 0.526i)T \)
29 \( 1 + (-0.445 + 0.895i)T \)
31 \( 1 + (-0.361 + 0.932i)T \)
37 \( 1 + (-0.183 + 0.982i)T \)
41 \( 1 + (-0.932 + 0.361i)T \)
43 \( 1 + (-0.895 + 0.445i)T \)
47 \( 1 + (-0.982 - 0.183i)T \)
53 \( 1 + (-0.526 + 0.850i)T \)
59 \( 1 + (-0.183 + 0.982i)T \)
61 \( 1 + (0.982 + 0.183i)T \)
67 \( 1 + T \)
71 \( 1 + (0.932 + 0.361i)T \)
73 \( 1 + (-0.982 + 0.183i)T \)
79 \( 1 + (-0.0922 - 0.995i)T \)
83 \( 1 + (0.982 + 0.183i)T \)
89 \( 1 + (-0.273 + 0.961i)T \)
97 \( 1 + (0.895 - 0.445i)T \)
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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

−20.815455706034483518535849464784, −20.272556490370479923824996677243, −19.64360965194744418371127561280, −18.87894139710237614893696346670, −18.15238573524719196603211245749, −17.201106157349222530153970148634, −15.853896472645189180854371292002, −15.15653697745413447432580158069, −14.48399342402872855496420296755, −13.81827949773966602987108485841, −13.06004230014217313096715227930, −12.6068965422547992346696935981, −11.496690373493599353604275459295, −10.18935223790068910625474832364, −9.92034619199877262819537551578, −9.3269236481496485260126810864, −7.71588278751054380931101991340, −6.940674534797595003171026453313, −6.27030113005589477450492406602, −5.12575934043081065117026825691, −3.95213461866160093444579428416, −3.34775823022167539759337147335, −2.23647061702495316788350522356, −1.94018379416572134467722982425, −0.2235817043437783499409834292, 1.783354696481538238252510123923, 2.64762876363134678510005868273, 3.459584377964780001954481855, 4.49278235065906912183834577977, 5.24149553379079440167419795319, 6.39296642871138119745259451812, 6.84605422656794524819841975609, 8.291857050777870894743413716169, 8.74617443665165416679372693148, 9.45133054513299607216726584624, 10.49281245757479167549141601226, 11.81660099492456764276144350619, 12.77075931114988351140374661060, 13.210870752362571030335364536298, 13.90969193097075231386415539662, 14.66666402684110954344863892079, 15.47165538805635963012390707655, 16.3238847755400159363304444173, 16.69425185687551147572360544678, 17.796086750676932853945442194559, 18.82421206812727210306779856485, 19.817199470891266310232067423088, 20.235575436786191179775180462694, 21.49831499258958188961989424890, 21.75989002445189313403610599916

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