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Properties

Label 1-4008-4008.581-r0-0-0
Degree $1$
Conductor $4008$
Sign $-0.541 - 0.840i$
Analytic cond. $18.6130$
Root an. cond. $18.6130$
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  + (0.982 − 0.188i)5-s + (−0.942 + 0.334i)7-s + (0.553 + 0.832i)11-s + (−0.800 + 0.599i)13-s + (0.822 − 0.569i)17-s + (0.0189 − 0.999i)19-s + (−0.993 − 0.113i)23-s + (0.929 − 0.369i)25-s + (−0.993 + 0.113i)29-s + (−0.243 + 0.969i)31-s + (−0.862 + 0.505i)35-s + (−0.521 − 0.853i)37-s + (−0.881 − 0.472i)41-s + (0.726 − 0.686i)43-s + (−0.988 − 0.150i)47-s + ⋯
L(s)  = 1  + (0.982 − 0.188i)5-s + (−0.942 + 0.334i)7-s + (0.553 + 0.832i)11-s + (−0.800 + 0.599i)13-s + (0.822 − 0.569i)17-s + (0.0189 − 0.999i)19-s + (−0.993 − 0.113i)23-s + (0.929 − 0.369i)25-s + (−0.993 + 0.113i)29-s + (−0.243 + 0.969i)31-s + (−0.862 + 0.505i)35-s + (−0.521 − 0.853i)37-s + (−0.881 − 0.472i)41-s + (0.726 − 0.686i)43-s + (−0.988 − 0.150i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.541 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.541 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4008\)    =    \(2^{3} \cdot 3 \cdot 167\)
Sign: $-0.541 - 0.840i$
Analytic conductor: \(18.6130\)
Root analytic conductor: \(18.6130\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4008} (581, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4008,\ (0:\ ),\ -0.541 - 0.840i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3414827391 - 0.6261004554i\)
\(L(\frac12)\) \(\approx\) \(0.3414827391 - 0.6261004554i\)
\(L(1)\) \(\approx\) \(0.9539863257 - 0.04466724110i\)
\(L(1)\) \(\approx\) \(0.9539863257 - 0.04466724110i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
167 \( 1 \)
good5 \( 1 + (0.982 - 0.188i)T \)
7 \( 1 + (-0.942 + 0.334i)T \)
11 \( 1 + (0.553 + 0.832i)T \)
13 \( 1 + (-0.800 + 0.599i)T \)
17 \( 1 + (0.822 - 0.569i)T \)
19 \( 1 + (0.0189 - 0.999i)T \)
23 \( 1 + (-0.993 - 0.113i)T \)
29 \( 1 + (-0.993 + 0.113i)T \)
31 \( 1 + (-0.243 + 0.969i)T \)
37 \( 1 + (-0.521 - 0.853i)T \)
41 \( 1 + (-0.881 - 0.472i)T \)
43 \( 1 + (0.726 - 0.686i)T \)
47 \( 1 + (-0.988 - 0.150i)T \)
53 \( 1 + (-0.206 - 0.978i)T \)
59 \( 1 + (-0.822 - 0.569i)T \)
61 \( 1 + (-0.997 - 0.0756i)T \)
67 \( 1 + (-0.982 - 0.188i)T \)
71 \( 1 + (0.614 + 0.788i)T \)
73 \( 1 + (-0.421 + 0.906i)T \)
79 \( 1 + (0.965 + 0.261i)T \)
83 \( 1 + (-0.280 - 0.959i)T \)
89 \( 1 + (0.644 - 0.764i)T \)
97 \( 1 + (-0.243 - 0.969i)T \)
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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

−18.78187902802584737083831902435, −18.05602517371002965437662629711, −17.17032168167456914452735151337, −16.68702705868292683878760947948, −16.33707920550957750842046670737, −15.08244183178540015693677096197, −14.690897876058658916814043170354, −13.71278194963538580264580480092, −13.4992818653951055299944916368, −12.4788981890237954095187752270, −12.1028345572083858360567643168, −10.93055897900373802006209596231, −10.33322191263780115544783875455, −9.6793055784099512078959314615, −9.32191068539036984341466898560, −8.15340798597649300741465347144, −7.59147895389134056500104805748, −6.56760892374420589838759536122, −5.94663674198405061682066972090, −5.62967203694937315278214396583, −4.42200119050720404113602214382, −3.447686530322913788779838375128, −3.04202024122573391789637713500, −1.93181491635153940199909249105, −1.16067367895108857109737432286, 0.187575282515610189804183364749, 1.60917672010298254342435471423, 2.17620405547216634068486224886, 3.02302206988351887234504545842, 3.908829167852409119756072076375, 4.93354091181236616396107330770, 5.42049757238301422076337947393, 6.3711159351717122155366582728, 6.906116523418101331908888977623, 7.55389945386115200666253656402, 8.90857581807803337842141933575, 9.22563421398483607356376977118, 9.87927385077468289716560715816, 10.364723897600964477804549280740, 11.51610216058645498591364876044, 12.284818444594023591712975639323, 12.63508532155370859739280717109, 13.49947588612665033149550330855, 14.18967943910170069290005935042, 14.67811929519735218703834726758, 15.6366200454789664852024427839, 16.27290551228703945336881925482, 16.95657741353953168958817181515, 17.48641397544487349768947748645, 18.23359041644949958239811155770

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