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 − 4·11-s − 2·17-s + 6·19-s + 6·23-s − 2·31-s − 2·37-s + 2·41-s + 4·43-s + 8·47-s + 49-s − 10·53-s − 4·59-s − 2·61-s + 12·67-s − 8·71-s − 8·73-s + 4·77-s + 8·79-s − 4·83-s + 10·89-s + 4·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 0.377·7-s − 1.20·11-s − 0.485·17-s + 1.37·19-s + 1.25·23-s − 0.359·31-s − 0.328·37-s + 0.312·41-s + 0.609·43-s + 1.16·47-s + 1/7·49-s − 1.37·53-s − 0.520·59-s − 0.256·61-s + 1.46·67-s − 0.949·71-s − 0.936·73-s + 0.455·77-s + 0.900·79-s − 0.439·83-s + 1.05·89-s + 0.406·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-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

\(L(1)\) \(\approx\) \(1.616773406\)
\(L(\frac12)\) \(\approx\) \(1.616773406\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
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 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−15.52171343443491, −14.90211353382826, −14.19390174942691, −13.73604171761683, −13.16587361461780, −12.76128959959616, −12.22195142053175, −11.51172117931090, −10.96056489693925, −10.52932389377326, −9.912687864687191, −9.203618644577686, −8.963077060328740, −8.002900621984103, −7.589338391953530, −7.044564378768907, −6.376117509031266, −5.584522651935406, −5.185826849980712, −4.529588763905436, −3.659085694954895, −2.959434311199486, −2.506667138070075, −1.447382468815879, −0.5134782022222728, 0.5134782022222728, 1.447382468815879, 2.506667138070075, 2.959434311199486, 3.659085694954895, 4.529588763905436, 5.185826849980712, 5.584522651935406, 6.376117509031266, 7.044564378768907, 7.589338391953530, 8.002900621984103, 8.963077060328740, 9.203618644577686, 9.912687864687191, 10.52932389377326, 10.96056489693925, 11.51172117931090, 12.22195142053175, 12.76128959959616, 13.16587361461780, 13.73604171761683, 14.19390174942691, 14.90211353382826, 15.52171343443491

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