Properties

Degree $2$
Conductor $25200$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7-s − 5·11-s + 6·13-s − 17-s + 3·19-s + 6·29-s + 4·31-s − 8·37-s − 11·41-s − 8·43-s − 2·47-s + 49-s + 4·53-s + 4·59-s − 2·61-s + 9·67-s − 10·71-s + 7·73-s + 5·77-s + 2·79-s − 11·83-s + 11·89-s − 6·91-s + 10·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 0.377·7-s − 1.50·11-s + 1.66·13-s − 0.242·17-s + 0.688·19-s + 1.11·29-s + 0.718·31-s − 1.31·37-s − 1.71·41-s − 1.21·43-s − 0.291·47-s + 1/7·49-s + 0.549·53-s + 0.520·59-s − 0.256·61-s + 1.09·67-s − 1.18·71-s + 0.819·73-s + 0.569·77-s + 0.225·79-s − 1.20·83-s + 1.16·89-s − 0.628·91-s + 1.01·97-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: \(1\)
Selberg data: \((2,\ 25200,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 5 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 - 3 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 11 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 9 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 + 11 T + p T^{2} \)
89 \( 1 - 11 T + p T^{2} \)
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

−15.67347624460527, −15.36700390018468, −14.55040291762610, −13.71127156373069, −13.52974587984491, −13.15912168573330, −12.40451125499426, −11.83247587468050, −11.32118378634922, −10.58326175459065, −10.27145774033011, −9.788620185553997, −8.825818514487468, −8.458047237762076, −8.032217489712845, −7.215651883811228, −6.593301607725541, −6.136457866000673, −5.215612589771834, −5.042289066518031, −3.978310892829260, −3.330175305033121, −2.839938288236029, −1.898041192524192, −1.031164935458281, 0, 1.031164935458281, 1.898041192524192, 2.839938288236029, 3.330175305033121, 3.978310892829260, 5.042289066518031, 5.215612589771834, 6.136457866000673, 6.593301607725541, 7.215651883811228, 8.032217489712845, 8.458047237762076, 8.825818514487468, 9.788620185553997, 10.27145774033011, 10.58326175459065, 11.32118378634922, 11.83247587468050, 12.40451125499426, 13.15912168573330, 13.52974587984491, 13.71127156373069, 14.55040291762610, 15.36700390018468, 15.67347624460527

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