Properties

Degree $2$
Conductor $25200$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7-s − 5·11-s + 3·13-s − 17-s − 6·19-s − 6·23-s + 9·29-s + 4·31-s − 2·37-s + 4·41-s + 10·43-s + 47-s + 49-s + 4·53-s − 8·59-s − 8·61-s + 12·67-s + 8·71-s − 2·73-s + 5·77-s − 13·79-s + 4·83-s − 4·89-s − 3·91-s + 13·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.377·7-s − 1.50·11-s + 0.832·13-s − 0.242·17-s − 1.37·19-s − 1.25·23-s + 1.67·29-s + 0.718·31-s − 0.328·37-s + 0.624·41-s + 1.52·43-s + 0.145·47-s + 1/7·49-s + 0.549·53-s − 1.04·59-s − 1.02·61-s + 1.46·67-s + 0.949·71-s − 0.234·73-s + 0.569·77-s − 1.46·79-s + 0.439·83-s − 0.423·89-s − 0.314·91-s + 1.31·97-s + 0.0995·101-s + 0.0985·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25200\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{25200} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 25200,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 + T \)
good11 \( 1 + 5 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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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

−15.61187162115122, −15.46513164044447, −14.38917743094346, −14.02153076440917, −13.47418841846679, −12.90782262303802, −12.49863684810181, −11.98345504828982, −11.11372678941342, −10.71167924036601, −10.23312669967738, −9.792673910173722, −8.818088260481296, −8.519629443501857, −7.894066134323285, −7.385123841805355, −6.384952882634597, −6.212360122180329, −5.493526687529579, −4.637160004263754, −4.215755831590648, −3.367313963858577, −2.577021740042159, −2.136507482071734, −0.9289715244765153, 0, 0.9289715244765153, 2.136507482071734, 2.577021740042159, 3.367313963858577, 4.215755831590648, 4.637160004263754, 5.493526687529579, 6.212360122180329, 6.384952882634597, 7.385123841805355, 7.894066134323285, 8.519629443501857, 8.818088260481296, 9.792673910173722, 10.23312669967738, 10.71167924036601, 11.11372678941342, 11.98345504828982, 12.49863684810181, 12.90782262303802, 13.47418841846679, 14.02153076440917, 14.38917743094346, 15.46513164044447, 15.61187162115122

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