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 − 4·11-s + 2·13-s + 2·17-s − 4·19-s + 8·23-s + 2·29-s − 6·37-s + 6·41-s − 4·43-s + 49-s − 10·53-s + 12·59-s + 14·61-s − 12·67-s − 8·71-s − 10·73-s + 4·77-s − 16·79-s + 12·83-s − 10·89-s − 2·91-s − 2·97-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 + 1.66·23-s + 0.371·29-s − 0.986·37-s + 0.937·41-s − 0.609·43-s + 1/7·49-s − 1.37·53-s + 1.56·59-s + 1.79·61-s − 1.46·67-s − 0.949·71-s − 1.17·73-s + 0.455·77-s − 1.80·79-s + 1.31·83-s − 1.05·89-s − 0.209·91-s − 0.203·97-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: \(1\)
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 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
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 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−15.87709618736166, −14.97308298765713, −14.73502095191157, −13.98830159313032, −13.28025962941111, −13.00666395267991, −12.60574211808593, −11.83086057456092, −11.22812007670652, −10.66294624741489, −10.29384724197237, −9.676732385514483, −8.927629290309323, −8.481521067642972, −7.951867537552738, −7.113259851749868, −6.828047813432167, −5.913652710223517, −5.491664054160612, −4.779532628626300, −4.149896087433062, −3.204780363476423, −2.863888017711161, −1.943802809439683, −0.9992720238278111, 0, 0.9992720238278111, 1.943802809439683, 2.863888017711161, 3.204780363476423, 4.149896087433062, 4.779532628626300, 5.491664054160612, 5.913652710223517, 6.828047813432167, 7.113259851749868, 7.951867537552738, 8.481521067642972, 8.927629290309323, 9.676732385514483, 10.29384724197237, 10.66294624741489, 11.22812007670652, 11.83086057456092, 12.60574211808593, 13.00666395267991, 13.28025962941111, 13.98830159313032, 14.73502095191157, 14.97308298765713, 15.87709618736166

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