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 − 2·13-s − 6·17-s + 4·19-s + 6·29-s + 4·31-s − 2·37-s − 6·41-s + 8·43-s + 12·47-s + 49-s + 6·53-s − 12·59-s + 2·61-s + 8·67-s − 14·73-s + 16·79-s − 12·83-s − 6·89-s − 2·91-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 6·119-s + ⋯
L(s)  = 1  + 0.377·7-s − 0.554·13-s − 1.45·17-s + 0.917·19-s + 1.11·29-s + 0.718·31-s − 0.328·37-s − 0.937·41-s + 1.21·43-s + 1.75·47-s + 1/7·49-s + 0.824·53-s − 1.56·59-s + 0.256·61-s + 0.977·67-s − 1.63·73-s + 1.80·79-s − 1.31·83-s − 0.635·89-s − 0.209·91-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 0.550·119-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

\(L(1)\) \(\approx\) \(2.055701956\)
\(L(\frac12)\) \(\approx\) \(2.055701956\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−15.44686438267232, −14.90662714993845, −14.16033107128434, −13.72721099720697, −13.45783827153023, −12.41326716352049, −12.27730488738791, −11.56484261270734, −11.02963517251610, −10.50122885677085, −9.907778274775014, −9.303000506521243, −8.698021730027144, −8.272075359525889, −7.435244955358690, −7.072315781778887, −6.381375096561224, −5.705460775013739, −5.007940936262809, −4.481484455773250, −3.890329532980016, −2.860179306831640, −2.428385186438322, −1.491641331850200, −0.5786971928961220, 0.5786971928961220, 1.491641331850200, 2.428385186438322, 2.860179306831640, 3.890329532980016, 4.481484455773250, 5.007940936262809, 5.705460775013739, 6.381375096561224, 7.072315781778887, 7.435244955358690, 8.272075359525889, 8.698021730027144, 9.303000506521243, 9.907778274775014, 10.50122885677085, 11.02963517251610, 11.56484261270734, 12.27730488738791, 12.41326716352049, 13.45783827153023, 13.72721099720697, 14.16033107128434, 14.90662714993845, 15.44686438267232

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