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 + 3·13-s + 7·17-s + 6·19-s − 9·23-s + 3·29-s + 7·31-s + 10·37-s − 41-s + 13·43-s + 2·47-s + 49-s − 53-s + 11·59-s + 13·61-s − 8·71-s − 8·73-s + 4·77-s − 4·79-s − 7·83-s − 14·89-s − 3·91-s − 8·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 0.377·7-s − 1.20·11-s + 0.832·13-s + 1.69·17-s + 1.37·19-s − 1.87·23-s + 0.557·29-s + 1.25·31-s + 1.64·37-s − 0.156·41-s + 1.98·43-s + 0.291·47-s + 1/7·49-s − 0.137·53-s + 1.43·59-s + 1.66·61-s − 0.949·71-s − 0.936·73-s + 0.455·77-s − 0.450·79-s − 0.768·83-s − 1.48·89-s − 0.314·91-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-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

\(L(1)\) \(\approx\) \(2.374327817\)
\(L(\frac12)\) \(\approx\) \(2.374327817\)
\(L(\frac{3}{2})\) 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 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + T + p T^{2} \)
43 \( 1 - 13 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 - 11 T + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 7 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−15.66281535843106, −14.73953615405047, −14.14529493816541, −13.86594499344590, −13.20159681344070, −12.71542342669172, −12.07792986619434, −11.68074307145889, −11.04504840940248, −10.19320727801749, −10.01337677119283, −9.560692295017352, −8.592046377811698, −8.052584517779973, −7.710429027500788, −7.078750660415440, −6.127809367935415, −5.732725847638587, −5.320764668414383, −4.293942233322493, −3.808659194522917, −2.878255347211969, −2.592724457331729, −1.320415584177313, −0.6706792731569695, 0.6706792731569695, 1.320415584177313, 2.592724457331729, 2.878255347211969, 3.808659194522917, 4.293942233322493, 5.320764668414383, 5.732725847638587, 6.127809367935415, 7.078750660415440, 7.710429027500788, 8.052584517779973, 8.592046377811698, 9.560692295017352, 10.01337677119283, 10.19320727801749, 11.04504840940248, 11.68074307145889, 12.07792986619434, 12.71542342669172, 13.20159681344070, 13.86594499344590, 14.14529493816541, 14.73953615405047, 15.66281535843106

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