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 2

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7-s − 4·11-s − 5·13-s + 3·17-s + 2·19-s + 23-s − 9·29-s − 3·31-s − 10·37-s + 5·41-s − 9·43-s − 6·47-s + 49-s − 3·53-s − 3·59-s − 5·61-s − 8·67-s − 4·73-s + 4·77-s + 4·79-s + 9·83-s + 6·89-s + 5·91-s + 8·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 0.377·7-s − 1.20·11-s − 1.38·13-s + 0.727·17-s + 0.458·19-s + 0.208·23-s − 1.67·29-s − 0.538·31-s − 1.64·37-s + 0.780·41-s − 1.37·43-s − 0.875·47-s + 1/7·49-s − 0.412·53-s − 0.390·59-s − 0.640·61-s − 0.977·67-s − 0.468·73-s + 0.455·77-s + 0.450·79-s + 0.987·83-s + 0.635·89-s + 0.524·91-s + 0.812·97-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  =  \(2\)
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 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 + 9 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−14.30189317776080, −13.63226550594695, −13.24564230230717, −12.78155032156872, −12.26532969566700, −11.96955267320260, −11.23406116194883, −10.80505593884231, −10.09494726829919, −10.00383401803741, −9.280663145009759, −8.939185267825503, −8.103162603896594, −7.612054381845130, −7.384953625129262, −6.786700620939759, −6.035298732643023, −5.486522354248417, −5.031287886000856, −4.690037878581455, −3.594780583650967, −3.354675650783340, −2.618311853790928, −2.044422906554389, −1.324542910239506, 0, 0, 1.324542910239506, 2.044422906554389, 2.618311853790928, 3.354675650783340, 3.594780583650967, 4.690037878581455, 5.031287886000856, 5.486522354248417, 6.035298732643023, 6.786700620939759, 7.384953625129262, 7.612054381845130, 8.103162603896594, 8.939185267825503, 9.280663145009759, 10.00383401803741, 10.09494726829919, 10.80505593884231, 11.23406116194883, 11.96955267320260, 12.26532969566700, 12.78155032156872, 13.24564230230717, 13.63226550594695, 14.30189317776080

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