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Properties

Label 2-4000-100.91-c0-0-0
Degree $2$
Conductor $4000$
Sign $-0.0314 - 0.999i$
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  + (0.951 − 0.309i)3-s + 1.61i·7-s + (−1.30 + 0.951i)13-s + (−0.587 − 0.190i)19-s + (0.500 + 1.53i)21-s + (−0.587 + 0.809i)23-s + (−0.587 + 0.809i)27-s + (−0.190 − 0.587i)29-s + (0.951 + 0.309i)31-s + (−0.809 + 0.587i)37-s + (−0.951 + 1.30i)39-s − 0.618i·43-s + (1.53 − 0.5i)47-s − 1.61·49-s + (0.309 + 0.951i)53-s + ⋯
L(s)  = 1  + (0.951 − 0.309i)3-s + 1.61i·7-s + (−1.30 + 0.951i)13-s + (−0.587 − 0.190i)19-s + (0.500 + 1.53i)21-s + (−0.587 + 0.809i)23-s + (−0.587 + 0.809i)27-s + (−0.190 − 0.587i)29-s + (0.951 + 0.309i)31-s + (−0.809 + 0.587i)37-s + (−0.951 + 1.30i)39-s − 0.618i·43-s + (1.53 − 0.5i)47-s − 1.61·49-s + (0.309 + 0.951i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0314 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0314 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $-0.0314 - 0.999i$
Analytic conductor: \(1.99626\)
Root analytic conductor: \(1.41289\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (1951, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4000,\ (\ :0),\ -0.0314 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.349230309\)
\(L(\frac12)\) \(\approx\) \(1.349230309\)
\(L(1)\) 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 \( 1 + (-0.951 + 0.309i)T + (0.809 - 0.587i)T^{2} \)
7 \( 1 - 1.61iT - T^{2} \)
11 \( 1 + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \)
17 \( 1 + (-0.809 - 0.587i)T^{2} \)
19 \( 1 + (0.587 + 0.190i)T + (0.809 + 0.587i)T^{2} \)
23 \( 1 + (0.587 - 0.809i)T + (-0.309 - 0.951i)T^{2} \)
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 \( 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2} \)
41 \( 1 + (0.309 - 0.951i)T^{2} \)
43 \( 1 + 0.618iT - T^{2} \)
47 \( 1 + (-1.53 + 0.5i)T + (0.809 - 0.587i)T^{2} \)
53 \( 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2} \)
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 \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \)
79 \( 1 + (0.587 - 0.190i)T + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (-0.951 - 0.309i)T + (0.809 + 0.587i)T^{2} \)
89 \( 1 + (0.309 + 0.951i)T^{2} \)
97 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
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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

−8.746491133000433485963573418523, −8.253595338696294372357407372546, −7.46124575027894671271054810777, −6.72845185826677989158284366430, −5.80147462905595997649701603856, −5.15902263361418326480020731209, −4.22706954287113224219582222954, −3.10860307334345410814658634908, −2.32371117744549810633837965577, −1.95519868848024717364140168301, 0.61141137689783561563809075641, 2.17167303544939574482900598708, 2.99886694531422314760539862061, 3.84792808343963239516333998619, 4.40461626678952350293642083608, 5.31796458978335664885111823196, 6.40662388691243263686420986219, 7.14483210249897194686944817386, 7.87253386156096891443956916207, 8.241029378551049914888964238678

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