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 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7-s + 2·13-s − 6·17-s + 8·19-s + 6·29-s + 4·31-s − 10·37-s + 6·41-s + 4·43-s + 49-s + 6·53-s + 12·59-s + 10·61-s + 4·67-s + 12·71-s + 10·73-s − 8·79-s + 12·83-s + 6·89-s + 2·91-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 6·119-s + ⋯
L(s)  = 1  + 0.377·7-s + 0.554·13-s − 1.45·17-s + 1.83·19-s + 1.11·29-s + 0.718·31-s − 1.64·37-s + 0.937·41-s + 0.609·43-s + 1/7·49-s + 0.824·53-s + 1.56·59-s + 1.28·61-s + 0.488·67-s + 1.42·71-s + 1.17·73-s − 0.900·79-s + 1.31·83-s + 0.635·89-s + 0.209·91-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 0.550·119-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  =  \(0\)
Selberg data  =  \((2,\ 100800,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(3.437252845\)
\(L(\frac12)\)  \(\approx\)  \(3.437252845\)
\(L(\frac{3}{2})\)   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 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−13.72034005000020, −13.40946540132388, −12.79792550839560, −12.18934870429066, −11.78639111425988, −11.27091909306667, −10.94136561331743, −10.24951026146921, −9.879799813542370, −9.233602205554709, −8.681801316190183, −8.423656997091192, −7.703779888895508, −7.211650752449551, −6.660707658770153, −6.240431052219112, −5.416621215808470, −5.100825484711666, −4.492826947409687, −3.793808810733583, −3.370559186635375, −2.470453630491154, −2.119082647396861, −1.081965406257777, −0.6836987412878588, 0.6836987412878588, 1.081965406257777, 2.119082647396861, 2.470453630491154, 3.370559186635375, 3.793808810733583, 4.492826947409687, 5.100825484711666, 5.416621215808470, 6.240431052219112, 6.660707658770153, 7.211650752449551, 7.703779888895508, 8.423656997091192, 8.681801316190183, 9.233602205554709, 9.879799813542370, 10.24951026146921, 10.94136561331743, 11.27091909306667, 11.78639111425988, 12.18934870429066, 12.79792550839560, 13.40946540132388, 13.72034005000020

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