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Properties

Label 1-4021-4021.11-r1-0-0
Degree $1$
Conductor $4021$
Sign $0.998 - 0.0481i$
Analytic cond. $432.116$
Root an. cond. $432.116$
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  + (−0.656 − 0.754i)2-s + (0.980 + 0.198i)3-s + (−0.138 + 0.990i)4-s + (−0.311 + 0.950i)5-s + (−0.493 − 0.869i)6-s + (−0.951 − 0.309i)7-s + (0.838 − 0.545i)8-s + (0.921 + 0.389i)9-s + (0.921 − 0.388i)10-s + (−0.905 + 0.425i)11-s + (−0.332 + 0.943i)12-s + (−0.960 + 0.277i)13-s + (0.390 + 0.920i)14-s + (−0.494 + 0.869i)15-s + (−0.961 − 0.274i)16-s + (0.264 + 0.964i)17-s + ⋯
L(s)  = 1  + (−0.656 − 0.754i)2-s + (0.980 + 0.198i)3-s + (−0.138 + 0.990i)4-s + (−0.311 + 0.950i)5-s + (−0.493 − 0.869i)6-s + (−0.951 − 0.309i)7-s + (0.838 − 0.545i)8-s + (0.921 + 0.389i)9-s + (0.921 − 0.388i)10-s + (−0.905 + 0.425i)11-s + (−0.332 + 0.943i)12-s + (−0.960 + 0.277i)13-s + (0.390 + 0.920i)14-s + (−0.494 + 0.869i)15-s + (−0.961 − 0.274i)16-s + (0.264 + 0.964i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.998 - 0.0481i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.998 - 0.0481i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4021\)
Sign: $0.998 - 0.0481i$
Analytic conductor: \(432.116\)
Root analytic conductor: \(432.116\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4021} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4021,\ (1:\ ),\ 0.998 - 0.0481i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9723901029 + 0.02343352282i\)
\(L(\frac12)\) \(\approx\) \(0.9723901029 + 0.02343352282i\)
\(L(1)\) \(\approx\) \(0.7486457552 + 0.01064959615i\)
\(L(1)\) \(\approx\) \(0.7486457552 + 0.01064959615i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4021 \( 1 \)
good2 \( 1 + (-0.656 - 0.754i)T \)
3 \( 1 + (0.980 + 0.198i)T \)
5 \( 1 + (-0.311 + 0.950i)T \)
7 \( 1 + (-0.951 - 0.309i)T \)
11 \( 1 + (-0.905 + 0.425i)T \)
13 \( 1 + (-0.960 + 0.277i)T \)
17 \( 1 + (0.264 + 0.964i)T \)
19 \( 1 + (-0.569 - 0.821i)T \)
23 \( 1 + (0.724 + 0.689i)T \)
29 \( 1 + (-0.658 + 0.752i)T \)
31 \( 1 + (-0.471 - 0.881i)T \)
37 \( 1 + (-0.104 - 0.994i)T \)
41 \( 1 + (0.283 - 0.958i)T \)
43 \( 1 + (0.649 - 0.760i)T \)
47 \( 1 + (0.866 + 0.5i)T \)
53 \( 1 + (-0.709 - 0.704i)T \)
59 \( 1 + (0.863 + 0.504i)T \)
61 \( 1 + (0.294 + 0.955i)T \)
67 \( 1 + (0.792 - 0.610i)T \)
71 \( 1 + (-0.830 + 0.557i)T \)
73 \( 1 + (-0.923 + 0.383i)T \)
79 \( 1 + (-0.487 - 0.872i)T \)
83 \( 1 + (-0.550 - 0.834i)T \)
89 \( 1 + (-0.425 - 0.905i)T \)
97 \( 1 + (-0.832 + 0.554i)T \)
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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

−18.505195191496715352511556099354, −17.63805653578211554643950516313, −16.59599266011490332489282267453, −16.4198546832838557126140767719, −15.549735219761703734644868382831, −15.182532424222196819114830216521, −14.318648801923504867164563008309, −13.63078543449896113333524650009, −12.7775168238391318246464056922, −12.57712942664500863128188350218, −11.39056898060535652397222303932, −10.253226377978968144490675654, −9.76163491915534433088561855688, −9.16802083317273499685604467175, −8.49938877665819212143548126530, −7.925901386086416161959833144833, −7.336750991059170399167108215055, −6.55232734856407499624534832060, −5.60828003015649660578677844932, −4.96125403306894626943133416502, −4.119353713700822421444856931477, −2.99111361375904306646762814777, −2.383670826603959428695989217589, −1.25574758460454645641958978448, −0.40871101351182514264338521448, 0.333621359142712770407427106654, 1.77555363467506148064390791318, 2.51939294144216679879439609038, 2.915619416941785135060498558689, 3.847681088261521207438645869162, 4.19721195998143513233159702427, 5.49593002595874634799277498925, 6.88237741234393684967352479650, 7.31804221078315550829753867012, 7.7078800500768955760992621803, 8.81296528275649884991757481975, 9.33189002131066628288085083058, 10.08245427895592046076291076169, 10.5317312010239412381890136090, 11.0937763126089184472576734075, 12.21161308754471640933166125310, 12.943923027559603728124948401438, 13.21570070323834661010373580124, 14.25490632526534072839221972366, 14.91128154618157826538297478527, 15.60985179781697273561460672542, 16.16679056882205422783726676430, 17.15910770083853472097605051359, 17.68480751055156427929450147440, 18.74461086478156992261202936978

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