Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7-s − 4·11-s − 2·13-s − 6·17-s − 8·19-s − 6·29-s − 8·31-s + 2·37-s − 2·41-s − 4·43-s + 8·47-s + 49-s + 6·53-s − 6·61-s − 4·67-s − 8·71-s − 10·73-s + 4·77-s − 16·79-s − 8·83-s + 6·89-s + 2·91-s + 6·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  − 0.377·7-s − 1.20·11-s − 0.554·13-s − 1.45·17-s − 1.83·19-s − 1.11·29-s − 1.43·31-s + 0.328·37-s − 0.312·41-s − 0.609·43-s + 1.16·47-s + 1/7·49-s + 0.824·53-s − 0.768·61-s − 0.488·67-s − 0.949·71-s − 1.17·73-s + 0.455·77-s − 1.80·79-s − 0.878·83-s + 0.635·89-s + 0.209·91-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

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

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: \(2\)
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 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−15.89960522179933, −15.19652847474308, −15.00558375429185, −14.43133483960224, −13.46701886196311, −13.21446964501041, −12.81190277459308, −12.27805406503613, −11.46057209468137, −10.94850076625075, −10.48242312986782, −10.04891498422979, −9.161860134823984, −8.824345973958107, −8.247480300114612, −7.333373742027227, −7.182427397259113, −6.277704789907098, −5.781665312500630, −5.087084129118901, −4.369628947821559, −3.910770932654989, −2.878390043197332, −2.337733081975300, −1.716947658537234, 0, 0, 1.716947658537234, 2.337733081975300, 2.878390043197332, 3.910770932654989, 4.369628947821559, 5.087084129118901, 5.781665312500630, 6.277704789907098, 7.182427397259113, 7.333373742027227, 8.247480300114612, 8.824345973958107, 9.161860134823984, 10.04891498422979, 10.48242312986782, 10.94850076625075, 11.46057209468137, 12.27805406503613, 12.81190277459308, 13.21446964501041, 13.46701886196311, 14.43133483960224, 15.00558375429185, 15.19652847474308, 15.89960522179933

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