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  + 1.61i·7-s − 9-s − 0.618·11-s − 0.618i·13-s − 1.61·19-s + 0.618i·23-s − 1.61i·37-s − 1.61·41-s − 0.618i·47-s − 1.61·49-s + 1.61i·53-s − 1.61·59-s − 1.61i·63-s − 1.00i·77-s + 81-s + ⋯
L(s)  = 1  + 1.61i·7-s − 9-s − 0.618·11-s − 0.618i·13-s − 1.61·19-s + 0.618i·23-s − 1.61i·37-s − 1.61·41-s − 0.618i·47-s − 1.61·49-s + 1.61i·53-s − 1.61·59-s − 1.61i·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  =  \((2,\ 4000,\ (\ :0),\ -1)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.2591473508\)
\(L(\frac12)\)  \(\approx\)  \(0.2591473508\)
\(L(1)\)   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 \( 1 - 1.61iT - T^{2} \)
11 \( 1 + 0.618T + T^{2} \)
13 \( 1 + 0.618iT - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + 1.61T + T^{2} \)
23 \( 1 - 0.618iT - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + 1.61iT - T^{2} \)
41 \( 1 + 1.61T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + 0.618iT - T^{2} \)
53 \( 1 - 1.61iT - T^{2} \)
59 \( 1 + 1.61T + T^{2} \)
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 \( 1 + 0.618T + T^{2} \)
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

−8.875073035910759696810007381054, −8.374087024418579464490554268339, −7.75453841086324745601825885864, −6.64742423570386658909579677805, −5.75608960533784646845657061716, −5.57721357462453009047090832336, −4.62740817848954176014717518225, −3.39993831743869432911254259836, −2.63438398402300114061068444108, −1.97396195465770005998945690585, 0.13130051037305364865967451466, 1.63929935785592869824700844551, 2.76888523229911173567052149398, 3.66849357286705439055417409692, 4.48465535260203657476214622750, 5.08971323717827158839558591371, 6.35781353451169588056136001653, 6.62853417019519606750557215045, 7.59379287691075400151524391234, 8.281800302504254038594602143201

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