Properties

Degree $2$
Conductor $273600$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·7-s − 4·11-s − 2·13-s − 2·17-s + 19-s − 8·23-s + 6·29-s + 4·31-s − 10·37-s + 2·41-s + 12·43-s + 9·49-s − 6·53-s + 10·61-s − 4·67-s + 8·71-s − 2·73-s + 16·77-s − 12·79-s + 8·83-s − 6·89-s + 8·91-s − 18·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  − 1.51·7-s − 1.20·11-s − 0.554·13-s − 0.485·17-s + 0.229·19-s − 1.66·23-s + 1.11·29-s + 0.718·31-s − 1.64·37-s + 0.312·41-s + 1.82·43-s + 9/7·49-s − 0.824·53-s + 1.28·61-s − 0.488·67-s + 0.949·71-s − 0.234·73-s + 1.82·77-s − 1.35·79-s + 0.878·83-s − 0.635·89-s + 0.838·91-s − 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273600\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 19\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{273600} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 273600,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 \)
19 \( 1 - T \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−12.94624576537306, −12.47324383858811, −12.21694912790529, −11.84179210816458, −10.96934569036533, −10.66804170473876, −10.15687247816081, −9.777625686412279, −9.519429218043576, −8.822784327012924, −8.305052602778699, −7.915718709830974, −7.346354518096051, −6.775245393537437, −6.526342778500794, −5.793958764341156, −5.554206134513491, −4.869814307692713, −4.264434854489880, −3.820701303207493, −3.140746145017620, −2.583834689454918, −2.403554896174243, −1.457431918597241, −0.5071774255541214, 0, 0.5071774255541214, 1.457431918597241, 2.403554896174243, 2.583834689454918, 3.140746145017620, 3.820701303207493, 4.264434854489880, 4.869814307692713, 5.554206134513491, 5.793958764341156, 6.526342778500794, 6.775245393537437, 7.346354518096051, 7.915718709830974, 8.305052602778699, 8.822784327012924, 9.519429218043576, 9.777625686412279, 10.15687247816081, 10.66804170473876, 10.96934569036533, 11.84179210816458, 12.21694912790529, 12.47324383858811, 12.94624576537306

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