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Properties

Label 2-4000-20.19-c0-0-5
Degree $2$
Conductor $4000$
Sign $0.707 + 0.707i$
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  − 1.61·3-s + 7-s + 1.61·9-s − i·11-s + 1.61i·13-s − i·17-s − 1.61i·19-s − 1.61·21-s − 27-s + 29-s + 0.618i·31-s + 1.61i·33-s − 2.61i·39-s + 41-s − 43-s + ⋯
L(s)  = 1  − 1.61·3-s + 7-s + 1.61·9-s − i·11-s + 1.61i·13-s − i·17-s − 1.61i·19-s − 1.61·21-s − 27-s + 29-s + 0.618i·31-s + 1.61i·33-s − 2.61i·39-s + 41-s − 43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(1.99626\)
Root analytic conductor: \(1.41289\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (3999, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4000,\ (\ :0),\ 0.707 + 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7904678948\)
\(L(\frac12)\) \(\approx\) \(0.7904678948\)
\(L(1)\) 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 \( 1 + 1.61T + T^{2} \)
7 \( 1 - T + T^{2} \)
11 \( 1 + iT - T^{2} \)
13 \( 1 - 1.61iT - T^{2} \)
17 \( 1 + iT - T^{2} \)
19 \( 1 + 1.61iT - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T + T^{2} \)
31 \( 1 - 0.618iT - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T + T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 + 0.618T + T^{2} \)
53 \( 1 + 0.618iT - T^{2} \)
59 \( 1 - 0.618iT - T^{2} \)
61 \( 1 + 1.61T + T^{2} \)
67 \( 1 - 0.618T + T^{2} \)
71 \( 1 + iT - T^{2} \)
73 \( 1 + 0.618iT - T^{2} \)
79 \( 1 - iT - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + 0.618iT - T^{2} \)
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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

−8.654547955860649450697028317008, −7.61049651912846960021963812336, −6.78791915377205901158550345637, −6.43187419429404153206293303974, −5.47923043703536918048730434549, −4.70069323542054278298463675761, −4.54452971770920375037687192289, −3.05996549815422088108326301628, −1.77439454314597711757064741245, −0.68376958510417186764408494987, 1.07904556784875570062588343217, 1.98478286030508818142116350718, 3.48592615637549310945588258206, 4.54882166413855651252357203775, 4.96504976219949857300362653745, 5.90806520187116067001145941268, 6.12559588596155914759705362497, 7.28878903727813446718688598513, 7.894652818344594194177023213455, 8.452427230156941721161850141470

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