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Properties

Label 1-4021-4021.4001-r0-0-0
Degree $1$
Conductor $4021$
Sign $-0.604 - 0.796i$
Analytic cond. $18.6734$
Root an. cond. $18.6734$
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  + (−0.680 − 0.732i)2-s + (−0.938 + 0.345i)3-s + (−0.0733 + 0.997i)4-s + (−0.922 − 0.386i)5-s + (0.892 + 0.451i)6-s + (−0.809 + 0.587i)7-s + (0.780 − 0.625i)8-s + (0.760 − 0.649i)9-s + (0.344 + 0.938i)10-s + (0.897 + 0.440i)11-s + (−0.276 − 0.961i)12-s + (−0.388 + 0.921i)13-s + (0.981 + 0.192i)14-s + (0.999 + 0.0437i)15-s + (−0.989 − 0.146i)16-s + (0.955 + 0.295i)17-s + ⋯
L(s)  = 1  + (−0.680 − 0.732i)2-s + (−0.938 + 0.345i)3-s + (−0.0733 + 0.997i)4-s + (−0.922 − 0.386i)5-s + (0.892 + 0.451i)6-s + (−0.809 + 0.587i)7-s + (0.780 − 0.625i)8-s + (0.760 − 0.649i)9-s + (0.344 + 0.938i)10-s + (0.897 + 0.440i)11-s + (−0.276 − 0.961i)12-s + (−0.388 + 0.921i)13-s + (0.981 + 0.192i)14-s + (0.999 + 0.0437i)15-s + (−0.989 − 0.146i)16-s + (0.955 + 0.295i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.604 - 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.604 - 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4021\)
Sign: $-0.604 - 0.796i$
Analytic conductor: \(18.6734\)
Root analytic conductor: \(18.6734\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4021} (4001, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4021,\ (0:\ ),\ -0.604 - 0.796i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03213054146 - 0.06468509148i\)
\(L(\frac12)\) \(\approx\) \(0.03213054146 - 0.06468509148i\)
\(L(1)\) \(\approx\) \(0.3967031061 + 0.01383888958i\)
\(L(1)\) \(\approx\) \(0.3967031061 + 0.01383888958i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4021 \( 1 \)
good2 \( 1 + (-0.680 - 0.732i)T \)
3 \( 1 + (-0.938 + 0.345i)T \)
5 \( 1 + (-0.922 - 0.386i)T \)
7 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (0.897 + 0.440i)T \)
13 \( 1 + (-0.388 + 0.921i)T \)
17 \( 1 + (0.955 + 0.295i)T \)
19 \( 1 + (-0.547 + 0.836i)T \)
23 \( 1 + (-0.0920 - 0.995i)T \)
29 \( 1 + (-0.979 - 0.201i)T \)
31 \( 1 + (-0.729 + 0.684i)T \)
37 \( 1 + (-0.978 - 0.207i)T \)
41 \( 1 + (0.996 + 0.0811i)T \)
43 \( 1 + (0.727 - 0.686i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (0.285 - 0.958i)T \)
59 \( 1 + (-0.881 - 0.471i)T \)
61 \( 1 + (-0.863 + 0.504i)T \)
67 \( 1 + (-0.116 - 0.993i)T \)
71 \( 1 + (0.545 + 0.838i)T \)
73 \( 1 + (-0.926 + 0.375i)T \)
79 \( 1 + (-0.812 + 0.582i)T \)
83 \( 1 + (-0.914 - 0.403i)T \)
89 \( 1 + (0.897 + 0.440i)T \)
97 \( 1 + (-0.433 + 0.901i)T \)
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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

−18.75415471415370623449415349858, −17.90962178579169332504971466354, −17.237091905743754818241947247882, −16.71298816622995159590398340911, −16.19722348745181600675161238633, −15.49528647651323544138521084065, −14.90388895773900297973329168539, −14.03505468600454978550549474237, −13.258134842117915138365637163822, −12.4998755147318692381887234204, −11.67920716656780575592157714175, −11.01018582388512577933598757366, −10.52316886985315649647015933555, −9.715829956313985144328409502607, −9.033613227874693278199877290614, −7.9179212358853365335635321548, −7.3416532737492347586224146857, −7.04941327562921200799953526667, −6.08601018361442889765365353352, −5.65410157837245944236361804857, −4.62826198379861699242999925218, −3.834255678196360764949544649791, −2.917780260107286269038900838283, −1.51421393847811055704975374513, −0.64588041968581137348336985060, 0.05255236996850496868781072158, 1.23269167009354851239403107506, 2.00691398785033190517949739906, 3.28507031438824861669182512358, 3.9375705498944963451680917781, 4.38918397396960820352711670786, 5.427571808342585581733835453383, 6.404698943004236465818079347290, 7.036272413091736735963657563289, 7.78692384322135521831775049935, 8.86233850831955502873570138947, 9.22910808409971370840712398491, 9.97170369105373374336641099389, 10.66348307469783796952815732621, 11.45215598975972911774237880950, 12.027355980169118836485172613018, 12.59707982122239077105257135519, 12.67509180552103587492904921388, 14.26983240278266155309238819414, 14.96603145083851851112340331687, 15.93776504301888121739384445214, 16.384487235961313977784273350469, 16.8456370911477380603686841403, 17.40702041163830180038541249053, 18.50538550443429964552077933782

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