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Properties

Label 1-4021-4021.385-r0-0-0
Degree $1$
Conductor $4021$
Sign $-0.567 + 0.823i$
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  + (0.865 − 0.501i)2-s + (−0.840 − 0.541i)3-s + (0.497 − 0.867i)4-s + (0.291 + 0.956i)5-s + (−0.998 − 0.0468i)6-s + (−0.809 − 0.587i)7-s + (−0.00468 − 0.999i)8-s + (0.413 + 0.910i)9-s + (0.731 + 0.681i)10-s + (−0.191 + 0.981i)11-s + (−0.887 + 0.460i)12-s + (−0.116 − 0.993i)13-s + (−0.994 − 0.102i)14-s + (0.273 − 0.961i)15-s + (−0.505 − 0.862i)16-s + (−0.940 + 0.340i)17-s + ⋯
L(s)  = 1  + (0.865 − 0.501i)2-s + (−0.840 − 0.541i)3-s + (0.497 − 0.867i)4-s + (0.291 + 0.956i)5-s + (−0.998 − 0.0468i)6-s + (−0.809 − 0.587i)7-s + (−0.00468 − 0.999i)8-s + (0.413 + 0.910i)9-s + (0.731 + 0.681i)10-s + (−0.191 + 0.981i)11-s + (−0.887 + 0.460i)12-s + (−0.116 − 0.993i)13-s + (−0.994 − 0.102i)14-s + (0.273 − 0.961i)15-s + (−0.505 − 0.862i)16-s + (−0.940 + 0.340i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.567 + 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.567 + 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4021\)
Sign: $-0.567 + 0.823i$
Analytic conductor: \(18.6734\)
Root analytic conductor: \(18.6734\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4021} (385, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4021,\ (0:\ ),\ -0.567 + 0.823i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1171793636 - 0.2229499163i\)
\(L(\frac12)\) \(\approx\) \(-0.1171793636 - 0.2229499163i\)
\(L(1)\) \(\approx\) \(0.9328739932 - 0.4513050119i\)
\(L(1)\) \(\approx\) \(0.9328739932 - 0.4513050119i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4021 \( 1 \)
good2 \( 1 + (0.865 - 0.501i)T \)
3 \( 1 + (-0.840 - 0.541i)T \)
5 \( 1 + (0.291 + 0.956i)T \)
7 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (-0.191 + 0.981i)T \)
13 \( 1 + (-0.116 - 0.993i)T \)
17 \( 1 + (-0.940 + 0.340i)T \)
19 \( 1 + (0.991 + 0.130i)T \)
23 \( 1 + (0.464 - 0.885i)T \)
29 \( 1 + (-0.388 - 0.921i)T \)
31 \( 1 + (-0.489 + 0.872i)T \)
37 \( 1 + (0.309 - 0.951i)T \)
41 \( 1 + (-0.987 + 0.158i)T \)
43 \( 1 + (-0.154 + 0.988i)T \)
47 \( 1 + T \)
53 \( 1 + (-0.0234 - 0.999i)T \)
59 \( 1 + (0.825 - 0.564i)T \)
61 \( 1 + (-0.860 + 0.509i)T \)
67 \( 1 + (-0.698 + 0.715i)T \)
71 \( 1 + (0.974 + 0.223i)T \)
73 \( 1 + (0.988 + 0.149i)T \)
79 \( 1 + (-0.227 - 0.973i)T \)
83 \( 1 + (-0.963 - 0.268i)T \)
89 \( 1 + (-0.191 + 0.981i)T \)
97 \( 1 + (-0.643 - 0.765i)T \)
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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

−18.7039854084117648655179416785, −18.13524680061337923753016635983, −17.021133038617182559100869134669, −16.79660077437896799245444022781, −16.196882547924271550245219957951, −15.568420058302724196804406734586, −15.214864181012940778431160707964, −13.85766488026226057972328583554, −13.55745819977706351178825389796, −12.77616620245966093740886096178, −12.094341412636025669941793635566, −11.5637720351519291268932069601, −10.97850015480150803446735285756, −9.800513432904716990516630662287, −9.063437376540723894212780154783, −8.77215685890218812098077073707, −7.478226489883925255912062961126, −6.69440817606291891274335982515, −6.06495657703365067583046358943, −5.35599472583777919539462647970, −5.0431692495070727796874231108, −4.05609515515182010119960917986, −3.44934842175966641405979864746, −2.48800237876845705674770972287, −1.338945722321971059401522848824, 0.05749366897511964197925666995, 1.19773851931051464444358123782, 2.185265280788618954480580213678, 2.77511929857189662857234493827, 3.66762154977786754172623848008, 4.4985157195390728323193101496, 5.352252740303981802108509808877, 5.973196433516890394904639525720, 6.80318247579240461262770216225, 7.03598997585241059748557993952, 7.86276908723886532275975513316, 9.45979521506022729142512947644, 10.10485057418353549689835011042, 10.57579689715652023663141066048, 11.113652115137615810802966331380, 11.93195341404383356742370572187, 12.73241793972553238565353774891, 13.06917978652612299088997646578, 13.71074354622265969548730095971, 14.512017969260041091069348786876, 15.26411469483444489082490531558, 15.83101108149418380484937185923, 16.645152465797701815774335966243, 17.58243979853617637664119783552, 18.036544378864186078889038579899

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