Properties

Degree $2$
Conductor $4000$
Sign $1$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.618i·7-s − 9-s + 1.61·11-s + 1.61i·13-s + 0.618·19-s − 1.61i·23-s + 0.618i·37-s + 0.618·41-s + 1.61i·47-s + 0.618·49-s − 0.618i·53-s + 0.618·59-s + 0.618i·63-s − 1.00i·77-s + 81-s + ⋯
L(s)  = 1  − 0.618i·7-s − 9-s + 1.61·11-s + 1.61i·13-s + 0.618·19-s − 1.61i·23-s + 0.618i·37-s + 0.618·41-s + 1.61i·47-s + 0.618·49-s − 0.618i·53-s + 0.618·59-s + 0.618i·63-s − 1.00i·77-s + 81-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $1$
Motivic weight: \(0\)
Character: $\chi_{4000} (751, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4000,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.342745298\)
\(L(\frac12)\) \(\approx\) \(1.342745298\)
\(L(1)\) 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 \( 1 + T^{2} \)
7 \( 1 + 0.618iT - T^{2} \)
11 \( 1 - 1.61T + T^{2} \)
13 \( 1 - 1.61iT - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - 0.618T + T^{2} \)
23 \( 1 + 1.61iT - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - 0.618iT - T^{2} \)
41 \( 1 - 0.618T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - 1.61iT - T^{2} \)
53 \( 1 + 0.618iT - T^{2} \)
59 \( 1 - 0.618T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - 1.61T + T^{2} \)
97 \( 1 + T^{2} \)
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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

−8.801790051693210850385347406114, −7.948321786656307458706766535921, −6.93690430752401324142094483678, −6.55656765581435062934760234396, −5.83882136065378387837589732836, −4.62108847752277971524425784973, −4.16410299623851445200865800315, −3.25454378175731826934697389640, −2.16660993313762805158250279725, −1.04841541239011216516111618840, 1.00443465304627483002935169179, 2.27878941582784016870294192260, 3.27945918539258542339187992909, 3.79257508186621277055952131660, 5.15339221181815923094476856809, 5.65174714330096384006790224702, 6.23723442122574676473765063285, 7.26278168039049891627197832289, 7.907074565090729262639057398264, 8.768706667180122260647628614832

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