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Properties

Label 1-4840-4840.2059-r0-0-0
Degree $1$
Conductor $4840$
Sign $0.988 - 0.152i$
Analytic cond. $22.4768$
Root an. cond. $22.4768$
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  + (−0.309 + 0.951i)3-s + (0.921 + 0.389i)7-s + (−0.809 − 0.587i)9-s + (0.870 − 0.491i)13-s + (0.941 − 0.336i)17-s + (0.0285 − 0.999i)19-s + (−0.654 + 0.755i)21-s + (−0.654 − 0.755i)23-s + (0.809 − 0.587i)27-s + (−0.564 − 0.825i)29-s + (−0.198 + 0.980i)31-s + (−0.736 + 0.676i)37-s + (0.198 + 0.980i)39-s + (0.254 + 0.967i)41-s + (−0.142 − 0.989i)43-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)3-s + (0.921 + 0.389i)7-s + (−0.809 − 0.587i)9-s + (0.870 − 0.491i)13-s + (0.941 − 0.336i)17-s + (0.0285 − 0.999i)19-s + (−0.654 + 0.755i)21-s + (−0.654 − 0.755i)23-s + (0.809 − 0.587i)27-s + (−0.564 − 0.825i)29-s + (−0.198 + 0.980i)31-s + (−0.736 + 0.676i)37-s + (0.198 + 0.980i)39-s + (0.254 + 0.967i)41-s + (−0.142 − 0.989i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.152i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.152i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4840\)    =    \(2^{3} \cdot 5 \cdot 11^{2}\)
Sign: $0.988 - 0.152i$
Analytic conductor: \(22.4768\)
Root analytic conductor: \(22.4768\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4840} (2059, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4840,\ (0:\ ),\ 0.988 - 0.152i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.687019452 - 0.1294663857i\)
\(L(\frac12)\) \(\approx\) \(1.687019452 - 0.1294663857i\)
\(L(1)\) \(\approx\) \(1.074334792 + 0.1846985516i\)
\(L(1)\) \(\approx\) \(1.074334792 + 0.1846985516i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
11 \( 1 \)
good3 \( 1 + (-0.309 + 0.951i)T \)
7 \( 1 + (0.921 + 0.389i)T \)
13 \( 1 + (0.870 - 0.491i)T \)
17 \( 1 + (0.941 - 0.336i)T \)
19 \( 1 + (0.0285 - 0.999i)T \)
23 \( 1 + (-0.654 - 0.755i)T \)
29 \( 1 + (-0.564 - 0.825i)T \)
31 \( 1 + (-0.198 + 0.980i)T \)
37 \( 1 + (-0.736 + 0.676i)T \)
41 \( 1 + (0.254 + 0.967i)T \)
43 \( 1 + (-0.142 - 0.989i)T \)
47 \( 1 + (0.774 - 0.633i)T \)
53 \( 1 + (0.974 - 0.226i)T \)
59 \( 1 + (-0.254 + 0.967i)T \)
61 \( 1 + (-0.998 + 0.0570i)T \)
67 \( 1 + (-0.841 + 0.540i)T \)
71 \( 1 + (-0.610 + 0.791i)T \)
73 \( 1 + (0.516 - 0.856i)T \)
79 \( 1 + (0.696 - 0.717i)T \)
83 \( 1 + (0.0855 - 0.996i)T \)
89 \( 1 + (-0.959 - 0.281i)T \)
97 \( 1 + (0.466 - 0.884i)T \)
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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

−18.182758178601645019915258079563, −17.576181617623972808521620433101, −16.7504688576424129987635654127, −16.4885157279034625310216153145, −15.42368930835050711301239782553, −14.55052885010472934651549752025, −13.99630107832102976600367281014, −13.61790630888533150604670340032, −12.5868447172841982868900651638, −12.19090401888346367691980120463, −11.34017061916741223857560731899, −10.933255557951703446656269010044, −10.155131019368189831907584747119, −9.12918493151283212030339676744, −8.39614343983951779784001045233, −7.64178279163561409912292993800, −7.42088009292061492537381075800, −6.281911837147441650991163286754, −5.76877943731664339648937749278, −5.11803054335984821447860219092, −4.01060833637485005748596630674, −3.46849263941436919814780275630, −2.16780588428935375180893939790, −1.58972186191783040355789479878, −0.976722149331694268146453766158, 0.54158368197132719191717629855, 1.57096933075655481289194166678, 2.65875932657792348134886122896, 3.34695538335078331747541793919, 4.227580449450139099228032835796, 4.85091424978201875813977058385, 5.57389250095096535481333217589, 6.05253225087723565124059603587, 7.11409414254730994991104876771, 7.98896319485055864702888814248, 8.7169277322451733104847267033, 9.099210040364140737637386858704, 10.32428646930107657546478458542, 10.4028104503187975205493005259, 11.468187410425708020593353882287, 11.75575590243047768609821357481, 12.563196758213014895832179425315, 13.636656212064911682157557159151, 14.11732404815240374831284349355, 15.06102251716555506120522759305, 15.279080220565692959888463093847, 16.10500549583889317044916586080, 16.702984527963921704069500664225, 17.399892796817228037052093415894, 18.10328119050766296911040339527

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