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Properties

Label 1-4004-4004.647-r0-0-0
Degree $1$
Conductor $4004$
Sign $0.966 - 0.256i$
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  + (0.309 − 0.951i)3-s + (−0.104 − 0.994i)5-s + (−0.809 − 0.587i)9-s + (−0.978 − 0.207i)15-s + (0.104 + 0.994i)17-s + (−0.309 + 0.951i)19-s + (0.5 + 0.866i)23-s + (−0.978 + 0.207i)25-s + (−0.809 + 0.587i)27-s + (−0.978 − 0.207i)29-s + (0.104 − 0.994i)31-s + (−0.669 + 0.743i)37-s + (0.669 + 0.743i)41-s + (0.5 + 0.866i)43-s + (−0.5 + 0.866i)45-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)3-s + (−0.104 − 0.994i)5-s + (−0.809 − 0.587i)9-s + (−0.978 − 0.207i)15-s + (0.104 + 0.994i)17-s + (−0.309 + 0.951i)19-s + (0.5 + 0.866i)23-s + (−0.978 + 0.207i)25-s + (−0.809 + 0.587i)27-s + (−0.978 − 0.207i)29-s + (0.104 − 0.994i)31-s + (−0.669 + 0.743i)37-s + (0.669 + 0.743i)41-s + (0.5 + 0.866i)43-s + (−0.5 + 0.866i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
Sign: $0.966 - 0.256i$
Analytic conductor: \(18.5944\)
Root analytic conductor: \(18.5944\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4004} (647, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4004,\ (0:\ ),\ 0.966 - 0.256i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.421266699 - 0.1852291074i\)
\(L(\frac12)\) \(\approx\) \(1.421266699 - 0.1852291074i\)
\(L(1)\) \(\approx\) \(0.9942037218 - 0.3560159156i\)
\(L(1)\) \(\approx\) \(0.9942037218 - 0.3560159156i\)

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

−18.56237334957080886681166697909, −17.77490260372180152361859087618, −17.18781307331461363250048964670, −16.22165039169003021386003390930, −15.79083172921343071021920771550, −15.1169933519202029294372973497, −14.38622047841820232149972042783, −14.0935171278333346324043300657, −13.17507079551284599181761307778, −12.26723880091533988676574849104, −11.22149372007907546151696170127, −11.024268842154618208198921295295, −10.25281934421306172312097937190, −9.53446295103952479587758753527, −8.90492452815117824746860177336, −8.15543735820655846565402400711, −7.14980107463206547570863077311, −6.76769805319546403242320689329, −5.60056455811622713915378300963, −5.02977416188128514542875291736, −4.11343705962564837773330115947, −3.44296810882353268732518261447, −2.688960866441603704568688129, −2.12859487141574657856615999518, −0.45001875604232190054464420675, 0.88844255915798136251188422583, 1.60168437939538935227057079825, 2.268126119943982851174015595261, 3.46745240571556802906784342951, 4.00965799525559200090759627625, 5.10949087652573586393378568767, 5.834158049106534595765652704913, 6.4113510075549467637535114344, 7.472041129120969708699594638682, 8.00280393363082315741906300141, 8.531017018146760236829521392614, 9.34028026026957951656063221007, 9.95558105166424288808741974919, 11.14176075747489935165240460006, 11.695870305591212793474974191317, 12.48752719790935413931060803139, 12.97974126093902104515804298097, 13.422944372744063438406451926901, 14.34510984280763161579731137031, 14.96840326598638374579131676679, 15.6829668877311689685404469596, 16.7107962993422569766549723580, 17.062061429327798664937088888265, 17.69594601323000543180433862085, 18.62152902456987466943127381415

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