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Properties

Label 1-4004-4004.1283-r0-0-0
Degree $1$
Conductor $4004$
Sign $0.619 - 0.784i$
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  + (0.809 + 0.587i)3-s + (−0.978 + 0.207i)5-s + (0.309 + 0.951i)9-s + (−0.913 − 0.406i)15-s + (0.978 − 0.207i)17-s + (−0.809 − 0.587i)19-s + (0.5 − 0.866i)23-s + (0.913 − 0.406i)25-s + (−0.309 + 0.951i)27-s + (−0.913 − 0.406i)29-s + (0.978 + 0.207i)31-s + (−0.104 − 0.994i)37-s + (0.104 − 0.994i)41-s + (−0.5 + 0.866i)43-s + (−0.5 − 0.866i)45-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)3-s + (−0.978 + 0.207i)5-s + (0.309 + 0.951i)9-s + (−0.913 − 0.406i)15-s + (0.978 − 0.207i)17-s + (−0.809 − 0.587i)19-s + (0.5 − 0.866i)23-s + (0.913 − 0.406i)25-s + (−0.309 + 0.951i)27-s + (−0.913 − 0.406i)29-s + (0.978 + 0.207i)31-s + (−0.104 − 0.994i)37-s + (0.104 − 0.994i)41-s + (−0.5 + 0.866i)43-s + (−0.5 − 0.866i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.619 - 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.619 - 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
Sign: $0.619 - 0.784i$
Analytic conductor: \(18.5944\)
Root analytic conductor: \(18.5944\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4004} (1283, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4004,\ (0:\ ),\ 0.619 - 0.784i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.218658117 - 0.5907004265i\)
\(L(\frac12)\) \(\approx\) \(1.218658117 - 0.5907004265i\)
\(L(1)\) \(\approx\) \(1.091612957 + 0.1144385930i\)
\(L(1)\) \(\approx\) \(1.091612957 + 0.1144385930i\)

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

−18.890994133087570588872904023943, −18.116112564418130143738757630, −17.094916596574888063069952069492, −16.65355407566523384870789754600, −15.60949611631820585487973463522, −15.158297849791215037582066026330, −14.5729586191639171105667190071, −13.81124690698250349862184499259, −13.04339654814737400894672466879, −12.45285606883785439765492533344, −11.89711845509766286304598802248, −11.14309412889814224230611400493, −10.21998756247688082795574983930, −9.405732676165015713284585442040, −8.73195805897421721260180013137, −7.83441319386756918988595954017, −7.78558311254534641039642576652, −6.76251733574592699677036798255, −6.0546725402324662860058692918, −4.985135235752602624445780242063, −4.138320135416876276827157544857, −3.40943450899507637007457193310, −2.88620670494166223623112808099, −1.673570833475032146970034309261, −1.068279829789860239429073760840, 0.362514693698032747441683314521, 1.708175078473616968819027341873, 2.73626903680317043590372194756, 3.23227054807200236323839262832, 4.125344803736308063819110013867, 4.59006863448092107124606496571, 5.48657561533473877562202425979, 6.59532527883774844052651363362, 7.37639589009834890963428664193, 7.95255522622690350444363982872, 8.650153095469381148962163944, 9.23276290066184785101693654453, 10.160592810723312353607580009292, 10.72754409800873959957584698226, 11.38656277664155026744388298944, 12.26560105458481736215667490886, 12.89901050787051903964883827123, 13.7727486296775542583488540642, 14.44825565290155119867014266310, 15.10337430995490236997535319696, 15.42472512408886231541425501782, 16.40910830366070921615441480666, 16.654483704062955859395649039003, 17.74631532961784671655633216165, 18.68517974279547762827027083516

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