Properties

Degree $2$
Conductor $25200$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7-s − 5·11-s − 13-s + 3·17-s + 6·19-s + 6·23-s + 9·29-s − 6·37-s − 8·41-s + 6·43-s − 3·47-s + 49-s − 12·53-s + 8·59-s − 4·61-s − 4·67-s + 8·71-s − 10·73-s + 5·77-s + 3·79-s + 12·83-s + 16·89-s + 91-s − 7·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 0.377·7-s − 1.50·11-s − 0.277·13-s + 0.727·17-s + 1.37·19-s + 1.25·23-s + 1.67·29-s − 0.986·37-s − 1.24·41-s + 0.914·43-s − 0.437·47-s + 1/7·49-s − 1.64·53-s + 1.04·59-s − 0.512·61-s − 0.488·67-s + 0.949·71-s − 1.17·73-s + 0.569·77-s + 0.337·79-s + 1.31·83-s + 1.69·89-s + 0.104·91-s − 0.710·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-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: \(0\)
Selberg data: \((2,\ 25200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.770680619\)
\(L(\frac12)\) \(\approx\) \(1.770680619\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 3 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 16 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−15.55055519395035, −14.84908252267391, −14.21412134632405, −13.74604457509140, −13.19422297591257, −12.74339989713207, −12.06136403779937, −11.77070273918895, −10.82080241817235, −10.48980720352163, −9.940539189115425, −9.403778445929913, −8.753990480050841, −8.011207295559557, −7.665308106261814, −6.981248644955956, −6.432839982493328, −5.540685292663391, −5.104504721107208, −4.695092155201445, −3.489478575719230, −3.072206370497234, −2.509396087950467, −1.410138384852940, −0.5469531897153716, 0.5469531897153716, 1.410138384852940, 2.509396087950467, 3.072206370497234, 3.489478575719230, 4.695092155201445, 5.104504721107208, 5.540685292663391, 6.432839982493328, 6.981248644955956, 7.665308106261814, 8.011207295559557, 8.753990480050841, 9.403778445929913, 9.940539189115425, 10.48980720352163, 10.82080241817235, 11.77070273918895, 12.06136403779937, 12.74339989713207, 13.19422297591257, 13.74604457509140, 14.21412134632405, 14.84908252267391, 15.55055519395035

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