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Properties

Label 1-4004-4004.103-r0-0-0
Degree $1$
Conductor $4004$
Sign $-0.865 - 0.501i$
Analytic cond. $18.5944$
Root an. cond. $18.5944$
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  + (0.913 − 0.406i)3-s + (−0.978 + 0.207i)5-s + (0.669 − 0.743i)9-s + (−0.809 + 0.587i)15-s + (−0.669 − 0.743i)17-s + (0.104 − 0.994i)19-s + (0.5 + 0.866i)23-s + (0.913 − 0.406i)25-s + (0.309 − 0.951i)27-s + (−0.809 + 0.587i)29-s + (0.978 + 0.207i)31-s + (−0.913 − 0.406i)37-s + (−0.809 − 0.587i)41-s − 43-s + (−0.5 + 0.866i)45-s + ⋯
L(s)  = 1  + (0.913 − 0.406i)3-s + (−0.978 + 0.207i)5-s + (0.669 − 0.743i)9-s + (−0.809 + 0.587i)15-s + (−0.669 − 0.743i)17-s + (0.104 − 0.994i)19-s + (0.5 + 0.866i)23-s + (0.913 − 0.406i)25-s + (0.309 − 0.951i)27-s + (−0.809 + 0.587i)29-s + (0.978 + 0.207i)31-s + (−0.913 − 0.406i)37-s + (−0.809 − 0.587i)41-s − 43-s + (−0.5 + 0.866i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.865 - 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.865 - 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
Sign: $-0.865 - 0.501i$
Analytic conductor: \(18.5944\)
Root analytic conductor: \(18.5944\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4004} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4004,\ (0:\ ),\ -0.865 - 0.501i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2634859177 - 0.9799254753i\)
\(L(\frac12)\) \(\approx\) \(0.2634859177 - 0.9799254753i\)
\(L(1)\) \(\approx\) \(1.025051004 - 0.2806765754i\)
\(L(1)\) \(\approx\) \(1.025051004 - 0.2806765754i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good3 \( 1 + (0.913 - 0.406i)T \)
5 \( 1 + (-0.978 + 0.207i)T \)
17 \( 1 + (-0.669 - 0.743i)T \)
19 \( 1 + (0.104 - 0.994i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.809 + 0.587i)T \)
31 \( 1 + (0.978 + 0.207i)T \)
37 \( 1 + (-0.913 - 0.406i)T \)
41 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 - T \)
47 \( 1 + (0.104 - 0.994i)T \)
53 \( 1 + (-0.978 - 0.207i)T \)
59 \( 1 + (0.104 + 0.994i)T \)
61 \( 1 + (0.978 - 0.207i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + (-0.104 - 0.994i)T \)
79 \( 1 + (-0.669 + 0.743i)T \)
83 \( 1 + (-0.309 + 0.951i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (0.309 + 0.951i)T \)
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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

−18.97059622990696532169844868391, −18.392392186531967647949575116, −17.13170712153326930306622963894, −16.7270431812467357668488456633, −15.69572687699905829744439084014, −15.54091206195007901377656712407, −14.67273738471627115992412065423, −14.23742633820705067528047319959, −13.174699326072561880931663044094, −12.80216912785192442125006838784, −11.88114564817736142033573829915, −11.18043923416037051573056812151, −10.3877970010785305440919274768, −9.79243461341220518661495060179, −8.8343708390637159187723751785, −8.33487255484608667756522735462, −7.847362163165647756175739759149, −6.99208794228526410020137637287, −6.18807141560521453166882343232, −4.9766738345999527899999874787, −4.43940158766063907306946483614, −3.69508806507885595100151133391, −3.123991469685674458075533730300, −2.13783453037826614251187134764, −1.2633713085858628887971631800, 0.24990915692674119255446925992, 1.36034223168233520103500242033, 2.33603087058789343106651993731, 3.13485310586752540325165087596, 3.64693070685849693823525795685, 4.56512687151863691578871054718, 5.27800932044148591425838145026, 6.656723933993860984849181257948, 7.05470751544081805811157518413, 7.61486744233652484919911121487, 8.56079684300611016416810883996, 8.90500191161644950151396500390, 9.75911991847457412587246389988, 10.651191427783842539962931663210, 11.50395837687160554318064853287, 11.93592306739758558273731798128, 12.86067493926692168389947334475, 13.452988285722097390691031838434, 14.05798864172822956005762696344, 14.907958181644249599091081443207, 15.46443248957251688479287417632, 15.81986482466250641042817343034, 16.8190078406918408660167789869, 17.74654716050991541153983816882, 18.28589738672169664517499315352

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