Properties

Degree 2
Conductor $ 2^{5} \cdot 5^{3} $
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

\( d \)  =  \(2\)
\( N \)  =  \(4000\)    =    \(2^{5} \cdot 5^{3}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{4000} (751, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 4000,\ (\ :0),\ 1)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(1.342745298\)
\(L(\frac12)\)  \(\approx\)  \(1.342745298\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 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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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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