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 − 4·11-s − 13-s − 3·17-s + 6·19-s − 3·23-s − 3·29-s − 9·31-s − 6·37-s + 5·41-s + 9·43-s + 6·47-s + 49-s + 9·53-s + 13·59-s − 7·61-s + 8·67-s + 16·71-s + 8·73-s + 4·77-s − 12·79-s + 15·83-s + 14·89-s + 91-s + 8·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.377·7-s − 1.20·11-s − 0.277·13-s − 0.727·17-s + 1.37·19-s − 0.625·23-s − 0.557·29-s − 1.61·31-s − 0.986·37-s + 0.780·41-s + 1.37·43-s + 0.875·47-s + 1/7·49-s + 1.23·53-s + 1.69·59-s − 0.896·61-s + 0.977·67-s + 1.89·71-s + 0.936·73-s + 0.455·77-s − 1.35·79-s + 1.64·83-s + 1.48·89-s + 0.104·91-s + 0.812·97-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 \( 1 + 4 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 9 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 - 9 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 13 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 15 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−15.74426425775202, −15.22886684024216, −14.46524520367265, −14.03549487683626, −13.41695318500237, −12.98922423274868, −12.44562182626307, −11.92840164632429, −11.21245312225062, −10.71603554346688, −10.26454832839482, −9.482188791018422, −9.205578244347927, −8.456330265662159, −7.676830269350161, −7.399024271392037, −6.765500741164068, −5.887688473654076, −5.394250189841212, −4.965408334500852, −3.897856792661050, −3.556960742199427, −2.468171032960883, −2.207071355309375, −0.9188802747435645, 0, 0.9188802747435645, 2.207071355309375, 2.468171032960883, 3.556960742199427, 3.897856792661050, 4.965408334500852, 5.394250189841212, 5.887688473654076, 6.765500741164068, 7.399024271392037, 7.676830269350161, 8.456330265662159, 9.205578244347927, 9.482188791018422, 10.26454832839482, 10.71603554346688, 11.21245312225062, 11.92840164632429, 12.44562182626307, 12.98922423274868, 13.41695318500237, 14.03549487683626, 14.46524520367265, 15.22886684024216, 15.74426425775202

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