Properties

Degree $2$
Conductor $4200$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 7-s + 9-s − 4·11-s + 2·13-s − 2·17-s − 4·19-s + 21-s + 27-s − 10·29-s − 4·33-s − 6·37-s + 2·39-s − 6·41-s + 4·43-s + 8·47-s + 49-s − 2·51-s − 6·53-s − 4·57-s − 4·59-s − 10·61-s + 63-s − 4·67-s − 16·71-s + 14·73-s − 4·77-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 1/3·9-s − 1.20·11-s + 0.554·13-s − 0.485·17-s − 0.917·19-s + 0.218·21-s + 0.192·27-s − 1.85·29-s − 0.696·33-s − 0.986·37-s + 0.320·39-s − 0.937·41-s + 0.609·43-s + 1.16·47-s + 1/7·49-s − 0.280·51-s − 0.824·53-s − 0.529·57-s − 0.520·59-s − 1.28·61-s + 0.125·63-s − 0.488·67-s − 1.89·71-s + 1.63·73-s − 0.455·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4200\)    =    \(2^{3} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{4200} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4200,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 \)
7 \( 1 - T \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.38919967144499, −17.85807153070219, −17.10118066393962, −16.50913029534749, −15.68777698530462, −15.27075318620278, −14.80235562916328, −13.84822305306759, −13.51207304852346, −12.82755050040280, −12.27563817269108, −11.27251355592484, −10.72679660351607, −10.27254064718307, −9.189673362045382, −8.830698235321685, −7.949477552936664, −7.561522870044442, −6.641176260713096, −5.780305497433333, −5.020570701397173, −4.183710882705429, −3.371497438635288, −2.398910065696213, −1.651070748888863, 0, 1.651070748888863, 2.398910065696213, 3.371497438635288, 4.183710882705429, 5.020570701397173, 5.780305497433333, 6.641176260713096, 7.561522870044442, 7.949477552936664, 8.830698235321685, 9.189673362045382, 10.27254064718307, 10.72679660351607, 11.27251355592484, 12.27563817269108, 12.82755050040280, 13.51207304852346, 13.84822305306759, 14.80235562916328, 15.27075318620278, 15.68777698530462, 16.50913029534749, 17.10118066393962, 17.85807153070219, 18.38919967144499

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