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Properties

Label 1-4004-4004.2511-r1-0-0
Degree $1$
Conductor $4004$
Sign $0.277 + 0.960i$
Analytic cond. $430.289$
Root an. cond. $430.289$
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 + (0.743 + 0.669i)5-s + (0.669 + 0.743i)9-s + (0.406 + 0.913i)15-s + (0.309 + 0.951i)17-s + (−0.994 + 0.104i)19-s + 23-s + (0.104 + 0.994i)25-s + (0.309 + 0.951i)27-s + (0.913 − 0.406i)29-s + (0.743 − 0.669i)31-s + (0.587 − 0.809i)37-s + (0.994 − 0.104i)41-s + (−0.5 − 0.866i)43-s + i·45-s + ⋯
L(s)  = 1  + (0.913 + 0.406i)3-s + (0.743 + 0.669i)5-s + (0.669 + 0.743i)9-s + (0.406 + 0.913i)15-s + (0.309 + 0.951i)17-s + (−0.994 + 0.104i)19-s + 23-s + (0.104 + 0.994i)25-s + (0.309 + 0.951i)27-s + (0.913 − 0.406i)29-s + (0.743 − 0.669i)31-s + (0.587 − 0.809i)37-s + (0.994 − 0.104i)41-s + (−0.5 − 0.866i)43-s + i·45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
Sign: $0.277 + 0.960i$
Analytic conductor: \(430.289\)
Root analytic conductor: \(430.289\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4004} (2511, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4004,\ (1:\ ),\ 0.277 + 0.960i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.055143764 + 3.050110647i\)
\(L(\frac12)\) \(\approx\) \(4.055143764 + 3.050110647i\)
\(L(1)\) \(\approx\) \(1.777842893 + 0.6138306503i\)
\(L(1)\) \(\approx\) \(1.777842893 + 0.6138306503i\)

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

−18.02129223924056562138299863026, −17.7522233305667421049376263263, −16.74692619479057530739782347876, −16.22361380421556077447562439130, −15.326944008577197945622996867984, −14.64873960258962579353018894556, −14.02156990143247912836343345386, −13.363757380653112693804520782540, −12.881770019855608507797152295231, −12.221252959586161868618553945306, −11.40550043217624964439527900181, −10.29545963061637702777842679973, −9.77050771026745096158495293881, −9.03223538886173227702389199306, −8.49391766308216183589089666781, −7.87447431101061438789605476137, −6.757872277904272967536278106651, −6.500944539241751223246581709065, −5.25768157345778470916430455300, −4.72362428143085035485146519244, −3.77700032757979750078711239377, −2.75518401290160342187676144927, −2.331340103954208392016954534978, −1.22093037829558093507031542409, −0.75585027950609858039120067361, 0.89492915163404201048770093336, 2.000788441263139975559204569023, 2.46574419363209375191544050486, 3.30830871608180346799758522794, 4.05386420539605775243827036291, 4.83598438522009314840574409757, 5.831457774414052368407284872731, 6.47978864303927270364020384873, 7.30030804853032968426226420968, 8.09002917317145376962934202285, 8.747530220367604077653858721917, 9.49313200632210163699154467035, 10.13677861476199711626416778364, 10.66955704838750294401980142448, 11.33427955854224227940981653582, 12.60563040085588921184298910228, 13.0377163525650204229521963522, 13.85049970276514212313230836330, 14.37772168130255242124585211692, 15.0319229782609512444725439560, 15.43422038579247149831767947567, 16.43605767619312318933916973236, 17.13528078841345760218260411603, 17.715464575038290532164670024341, 18.651776550961763909406238244136

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