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

−13.07996561827630, −12.30047473925009, −12.13025590699145, −11.57456238520823, −11.18259797220070, −10.82723785863870, −10.36000923175387, −9.677306453418730, −9.284196571342833, −8.704020920307855, −8.412463266967034, −8.033398897338699, −7.211899922233409, −6.897086587667446, −6.644934148020107, −5.800239782180480, −5.164276530311820, −4.943110127129954, −4.422879433694834, −3.886927019470221, −3.253797827237839, −2.647275341221118, −1.894978020994035, −1.508213598450835, −0.9680686464382859, 0, 0.9680686464382859, 1.508213598450835, 1.894978020994035, 2.647275341221118, 3.253797827237839, 3.886927019470221, 4.422879433694834, 4.943110127129954, 5.164276530311820, 5.800239782180480, 6.644934148020107, 6.897086587667446, 7.211899922233409, 8.033398897338699, 8.412463266967034, 8.704020920307855, 9.284196571342833, 9.677306453418730, 10.36000923175387, 10.82723785863870, 11.18259797220070, 11.57456238520823, 12.13025590699145, 12.30047473925009, 13.07996561827630

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