Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7-s − 4·11-s − 6·13-s + 6·17-s + 4·19-s − 4·23-s − 2·29-s + 8·31-s + 6·37-s − 6·41-s + 8·43-s + 49-s − 6·53-s − 4·59-s − 10·61-s + 8·67-s − 12·71-s + 14·73-s + 4·77-s − 16·79-s + 12·83-s − 14·89-s + 6·91-s − 18·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 0.377·7-s − 1.20·11-s − 1.66·13-s + 1.45·17-s + 0.917·19-s − 0.834·23-s − 0.371·29-s + 1.43·31-s + 0.986·37-s − 0.937·41-s + 1.21·43-s + 1/7·49-s − 0.824·53-s − 0.520·59-s − 1.28·61-s + 0.977·67-s − 1.42·71-s + 1.63·73-s + 0.455·77-s − 1.80·79-s + 1.31·83-s − 1.48·89-s + 0.628·91-s − 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(100800\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{100800} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 100800,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + T \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−14.01083591175273, −13.60326031585679, −12.83599688423886, −12.58750234064186, −11.99203866513380, −11.79877492644573, −10.99017825462206, −10.43056958641640, −9.853324020289277, −9.804849831780719, −9.249466575345145, −8.305886182315028, −7.928727424622043, −7.506288953320082, −7.164922273872989, −6.304241407318697, −5.798643030279056, −5.297391673509524, −4.812032395685573, −4.259234301959912, −3.393845974615855, −2.846901914396984, −2.527613675713726, −1.638180467172831, −0.7577771949125422, 0, 0.7577771949125422, 1.638180467172831, 2.527613675713726, 2.846901914396984, 3.393845974615855, 4.259234301959912, 4.812032395685573, 5.297391673509524, 5.798643030279056, 6.304241407318697, 7.164922273872989, 7.506288953320082, 7.928727424622043, 8.305886182315028, 9.249466575345145, 9.804849831780719, 9.853324020289277, 10.43056958641640, 10.99017825462206, 11.79877492644573, 11.99203866513380, 12.58750234064186, 12.83599688423886, 13.60326031585679, 14.01083591175273

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