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 + 2·13-s + 2·17-s + 4·19-s + 10·29-s − 6·37-s + 6·41-s − 4·43-s + 8·47-s + 49-s + 6·53-s − 4·59-s − 10·61-s + 4·67-s − 16·71-s + 14·73-s + 4·77-s − 8·79-s + 4·83-s − 10·89-s − 2·91-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  − 0.377·7-s − 1.20·11-s + 0.554·13-s + 0.485·17-s + 0.917·19-s + 1.85·29-s − 0.986·37-s + 0.937·41-s − 0.609·43-s + 1.16·47-s + 1/7·49-s + 0.824·53-s − 0.520·59-s − 1.28·61-s + 0.488·67-s − 1.89·71-s + 1.63·73-s + 0.455·77-s − 0.900·79-s + 0.439·83-s − 1.05·89-s − 0.209·91-s − 1.01·97-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: \(0\)
Selberg data: \((2,\ 25200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.949309537\)
\(L(\frac12)\) \(\approx\) \(1.949309537\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−15.52343059547052, −14.93213124777185, −13.98550411251073, −13.88804782830481, −13.29705543841984, −12.56535861145189, −12.25900333266229, −11.62121403515443, −10.92999187363787, −10.35528692483773, −10.07564454317036, −9.336940239880399, −8.671336475978897, −8.213292435969936, −7.493784268583799, −7.107769489978587, −6.232615014409792, −5.763816484526672, −5.106630246478910, −4.526183720000020, −3.638287555091321, −3.011009549980097, −2.489962041156340, −1.403976542785034, −0.5790646714514220, 0.5790646714514220, 1.403976542785034, 2.489962041156340, 3.011009549980097, 3.638287555091321, 4.526183720000020, 5.106630246478910, 5.763816484526672, 6.232615014409792, 7.107769489978587, 7.493784268583799, 8.213292435969936, 8.671336475978897, 9.336940239880399, 10.07564454317036, 10.35528692483773, 10.92999187363787, 11.62121403515443, 12.25900333266229, 12.56535861145189, 13.29705543841984, 13.88804782830481, 13.98550411251073, 14.93213124777185, 15.52343059547052

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