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Properties

Label 1-4004-4004.1623-r0-0-0
Degree $1$
Conductor $4004$
Sign $-0.712 + 0.701i$
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  + (−0.978 − 0.207i)3-s + (−0.587 + 0.809i)5-s + (0.913 + 0.406i)9-s + (0.743 − 0.669i)15-s + (0.104 − 0.994i)17-s + (−0.743 − 0.669i)19-s + (−0.5 + 0.866i)23-s + (−0.309 − 0.951i)25-s + (−0.809 − 0.587i)27-s + (−0.669 − 0.743i)29-s + (−0.587 − 0.809i)31-s + (0.743 − 0.669i)37-s + (0.207 − 0.978i)41-s + (0.5 + 0.866i)43-s + (−0.866 + 0.5i)45-s + ⋯
L(s)  = 1  + (−0.978 − 0.207i)3-s + (−0.587 + 0.809i)5-s + (0.913 + 0.406i)9-s + (0.743 − 0.669i)15-s + (0.104 − 0.994i)17-s + (−0.743 − 0.669i)19-s + (−0.5 + 0.866i)23-s + (−0.309 − 0.951i)25-s + (−0.809 − 0.587i)27-s + (−0.669 − 0.743i)29-s + (−0.587 − 0.809i)31-s + (0.743 − 0.669i)37-s + (0.207 − 0.978i)41-s + (0.5 + 0.866i)43-s + (−0.866 + 0.5i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.712 + 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.712 + 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
Sign: $-0.712 + 0.701i$
Analytic conductor: \(18.5944\)
Root analytic conductor: \(18.5944\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4004} (1623, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4004,\ (0:\ ),\ -0.712 + 0.701i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1030636494 + 0.2517461559i\)
\(L(\frac12)\) \(\approx\) \(0.1030636494 + 0.2517461559i\)
\(L(1)\) \(\approx\) \(0.6077197093 + 0.02786528186i\)
\(L(1)\) \(\approx\) \(0.6077197093 + 0.02786528186i\)

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 \( 1 + (-0.978 - 0.207i)T \)
5 \( 1 + (-0.587 + 0.809i)T \)
17 \( 1 + (0.104 - 0.994i)T \)
19 \( 1 + (-0.743 - 0.669i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.669 - 0.743i)T \)
31 \( 1 + (-0.587 - 0.809i)T \)
37 \( 1 + (0.743 - 0.669i)T \)
41 \( 1 + (0.207 - 0.978i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (0.951 - 0.309i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (0.207 + 0.978i)T \)
61 \( 1 + (-0.104 + 0.994i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 + (0.994 + 0.104i)T \)
73 \( 1 + (-0.951 - 0.309i)T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (-0.587 + 0.809i)T \)
89 \( 1 + (-0.866 - 0.5i)T \)
97 \( 1 + (0.406 - 0.913i)T \)
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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

−18.28587619430751306976833629576, −17.118159187447034862467730943937, −17.06417329688108626289506738029, −16.201556648776795060609152959674, −15.76389616102910689465685758868, −14.91705323336645902199596347668, −14.308041061095523372942652564953, −13.01584428828678669757616855020, −12.65026689284644505806680966200, −12.18718419219938796844634978455, −11.289013210781184886025694091401, −10.76711640063677461387547972997, −10.040223170739120564749421181891, −9.23017254539630069219997143303, −8.41046322713027330076765022514, −7.82457946950640359272309032922, −6.86176073049474957745810946988, −6.14425838778471928371554029707, −5.46037883055607900807787571560, −4.68364143681997339651743554595, −4.080148223298526797742996729772, −3.41819785558027348211737570216, −1.94042314305776390871289029138, −1.2258069763552241911588236000, −0.11934579663862369096269528429, 0.83368595405489502705928835876, 2.09278349877099287549317665068, 2.770330893873517817916816594298, 4.00106751867929745283022129198, 4.33240106316314828973026158506, 5.53546437251604300883104645766, 5.95568531133196587892743755253, 6.946354302855486484520458066154, 7.35697513660034400377392369913, 7.99026270394597718976224720570, 9.18822250799066638106336547234, 9.85299629490999702672721714251, 10.70627096225452287606494675952, 11.26160244654735976293233296346, 11.66987299718728888751306841083, 12.45945955781327003710319793552, 13.20599695511061637118278547727, 13.915630385025170475012113840994, 14.75615987197340689621697012953, 15.53427240432549971097313823111, 15.919130726639049999618906600703, 16.77852510138951875395102092518, 17.4209936715635360853534975612, 18.11571540606592178953050894847, 18.618437552457058815614684494756

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