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Properties

Label 2-4000-100.39-c0-0-1
Degree $2$
Conductor $4000$
Sign $0.866 - 0.498i$
Analytic cond. $1.99626$
Root an. cond. $1.41289$
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 − 1.61·7-s + (0.951 − 1.30i)13-s + (0.587 − 0.190i)19-s + (0.500 − 1.53i)21-s + (0.809 − 0.587i)23-s + (−0.809 + 0.587i)27-s + (0.190 − 0.587i)29-s + (0.951 − 0.309i)31-s + (−0.587 + 0.809i)37-s + (0.951 + 1.30i)39-s − 0.618·43-s + (0.5 − 1.53i)47-s + 1.61·49-s + (0.951 + 0.309i)53-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)3-s − 1.61·7-s + (0.951 − 1.30i)13-s + (0.587 − 0.190i)19-s + (0.500 − 1.53i)21-s + (0.809 − 0.587i)23-s + (−0.809 + 0.587i)27-s + (0.190 − 0.587i)29-s + (0.951 − 0.309i)31-s + (−0.587 + 0.809i)37-s + (0.951 + 1.30i)39-s − 0.618·43-s + (0.5 − 1.53i)47-s + 1.61·49-s + (0.951 + 0.309i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $0.866 - 0.498i$
Analytic conductor: \(1.99626\)
Root analytic conductor: \(1.41289\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (3199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4000,\ (\ :0),\ 0.866 - 0.498i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.017006404\)
\(L(\frac12)\) \(\approx\) \(1.017006404\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \)
7 \( 1 + 1.61T + T^{2} \)
11 \( 1 + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (-0.951 + 1.30i)T + (-0.309 - 0.951i)T^{2} \)
17 \( 1 + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (-0.587 + 0.190i)T + (0.809 - 0.587i)T^{2} \)
23 \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \)
29 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
31 \( 1 + (-0.951 + 0.309i)T + (0.809 - 0.587i)T^{2} \)
37 \( 1 + (0.587 - 0.809i)T + (-0.309 - 0.951i)T^{2} \)
41 \( 1 + (0.309 + 0.951i)T^{2} \)
43 \( 1 + 0.618T + T^{2} \)
47 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
53 \( 1 + (-0.951 - 0.309i)T + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (0.363 - 0.5i)T + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2} \)
67 \( 1 + (-0.809 + 0.587i)T^{2} \)
71 \( 1 + (-1.53 - 0.5i)T + (0.809 + 0.587i)T^{2} \)
73 \( 1 + (-0.587 - 0.809i)T + (-0.309 + 0.951i)T^{2} \)
79 \( 1 + (-0.587 - 0.190i)T + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2} \)
89 \( 1 + (0.309 - 0.951i)T^{2} \)
97 \( 1 + (-1.53 - 0.5i)T + (0.809 + 0.587i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.821205876840035274754720959259, −8.076037178618674197506051354642, −7.06371006044053736514036709771, −6.41670088496322517880393766691, −5.65306874604024214801169087330, −5.02033992367019978968476922856, −3.97725741575412267391881203801, −3.37989098451508095254733239614, −2.64944933096877302932477146922, −0.820347105564426561840416847261, 0.939987320435293528483016134650, 1.95046209959734491881224361794, 3.19512390882343194869282526032, 3.74946982931191428360685380634, 4.87284374586964601181545062284, 6.03937371693810712889644318759, 6.36977103464190917014679987394, 6.99140859343676274991284029153, 7.54892974476590916513389388346, 8.658199058088188273451428666850

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