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Properties

Label 2-4000-100.71-c0-0-0
Degree $2$
Conductor $4000$
Sign $0.975 + 0.218i$
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  + (0.587 − 0.809i)3-s + 0.618i·7-s + (−0.190 − 0.587i)13-s + (0.951 + 1.30i)19-s + (0.500 + 0.363i)21-s + (0.951 + 0.309i)23-s + (0.951 + 0.309i)27-s + (−1.30 − 0.951i)29-s + (0.587 + 0.809i)31-s + (0.309 + 0.951i)37-s + (−0.587 − 0.190i)39-s − 1.61i·43-s + (−0.363 + 0.5i)47-s + 0.618·49-s + (−0.809 − 0.587i)53-s + ⋯
L(s)  = 1  + (0.587 − 0.809i)3-s + 0.618i·7-s + (−0.190 − 0.587i)13-s + (0.951 + 1.30i)19-s + (0.500 + 0.363i)21-s + (0.951 + 0.309i)23-s + (0.951 + 0.309i)27-s + (−1.30 − 0.951i)29-s + (0.587 + 0.809i)31-s + (0.309 + 0.951i)37-s + (−0.587 − 0.190i)39-s − 1.61i·43-s + (−0.363 + 0.5i)47-s + 0.618·49-s + (−0.809 − 0.587i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.218i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.218i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $0.975 + 0.218i$
Analytic conductor: \(1.99626\)
Root analytic conductor: \(1.41289\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4000,\ (\ :0),\ 0.975 + 0.218i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.668009202\)
\(L(\frac12)\) \(\approx\) \(1.668009202\)
\(L(1)\) 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 \( 1 + (-0.587 + 0.809i)T + (-0.309 - 0.951i)T^{2} \)
7 \( 1 - 0.618iT - T^{2} \)
11 \( 1 + (0.809 + 0.587i)T^{2} \)
13 \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \)
17 \( 1 + (0.309 - 0.951i)T^{2} \)
19 \( 1 + (-0.951 - 1.30i)T + (-0.309 + 0.951i)T^{2} \)
23 \( 1 + (-0.951 - 0.309i)T + (0.809 + 0.587i)T^{2} \)
29 \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \)
31 \( 1 + (-0.587 - 0.809i)T + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2} \)
41 \( 1 + (-0.809 + 0.587i)T^{2} \)
43 \( 1 + 1.61iT - T^{2} \)
47 \( 1 + (0.363 - 0.5i)T + (-0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (-1.53 + 0.5i)T + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2} \)
67 \( 1 + (-0.309 + 0.951i)T^{2} \)
71 \( 1 + (0.363 - 0.5i)T + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \)
79 \( 1 + (-0.951 + 1.30i)T + (-0.309 - 0.951i)T^{2} \)
83 \( 1 + (-0.587 - 0.809i)T + (-0.309 + 0.951i)T^{2} \)
89 \( 1 + (-0.809 - 0.587i)T^{2} \)
97 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
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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

−8.335575060393103160585484779401, −7.971363384617850138617679619225, −7.26255277641946837078865496262, −6.54842746729831029338842924625, −5.54700260766591206074971020827, −5.10431803344210262317614235275, −3.78895291126535814909057504812, −2.99076048455477971258066265724, −2.17063981732571658343598549230, −1.24862914838564092267605161105, 1.04979277853090957647010358349, 2.51048719334938894121845472683, 3.27191073317770775484774094574, 4.10281807801000477339729234415, 4.68479374609467529436934309056, 5.51509514324657924236651514937, 6.65830723025083279966346034966, 7.17654781808108292635280724594, 7.931867050469965439229663966808, 8.991635633925606676008438172500

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