Properties

Degree $2$
Conductor $235200$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s − 6·11-s + 4·13-s + 2·17-s + 4·19-s − 6·23-s − 27-s − 6·29-s + 6·33-s − 10·37-s − 4·39-s − 6·41-s + 8·43-s + 8·47-s − 2·51-s + 6·53-s − 4·57-s + 4·59-s + 4·61-s − 12·67-s + 6·69-s + 6·71-s − 16·73-s + 12·79-s + 81-s − 12·83-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 1.80·11-s + 1.10·13-s + 0.485·17-s + 0.917·19-s − 1.25·23-s − 0.192·27-s − 1.11·29-s + 1.04·33-s − 1.64·37-s − 0.640·39-s − 0.937·41-s + 1.21·43-s + 1.16·47-s − 0.280·51-s + 0.824·53-s − 0.529·57-s + 0.520·59-s + 0.512·61-s − 1.46·67-s + 0.722·69-s + 0.712·71-s − 1.87·73-s + 1.35·79-s + 1/9·81-s − 1.31·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(235200\)    =    \(2^{6} \cdot 3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{235200} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 235200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8250879893\)
\(L(\frac12)\) \(\approx\) \(0.8250879893\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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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

−12.89358240899764, −12.39428020248306, −11.99832199910985, −11.55822517486922, −10.92055334460124, −10.63706631268958, −10.24172007392174, −9.790740657522434, −9.229818464794758, −8.537502778352122, −8.247055575048877, −7.637731671167435, −7.259157962744076, −6.829643034284023, −5.913743621757536, −5.616225036742533, −5.487767663547076, −4.752754293840254, −4.086205121545955, −3.615319384090574, −3.044961768939768, −2.384724045206720, −1.754753951551885, −1.104540179572734, −0.2776178351144511, 0.2776178351144511, 1.104540179572734, 1.754753951551885, 2.384724045206720, 3.044961768939768, 3.615319384090574, 4.086205121545955, 4.752754293840254, 5.487767663547076, 5.616225036742533, 5.913743621757536, 6.829643034284023, 7.259157962744076, 7.637731671167435, 8.247055575048877, 8.537502778352122, 9.229818464794758, 9.790740657522434, 10.24172007392174, 10.63706631268958, 10.92055334460124, 11.55822517486922, 11.99832199910985, 12.39428020248306, 12.89358240899764

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