Properties

Degree $2$
Conductor $53900$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 9-s − 11-s − 4·13-s + 4·19-s + 6·23-s + 4·27-s − 6·29-s − 8·31-s + 2·33-s − 2·37-s + 8·39-s − 6·41-s − 8·43-s + 6·47-s + 6·53-s − 8·57-s + 12·59-s − 2·61-s + 10·67-s − 12·69-s − 12·71-s − 16·73-s + 8·79-s − 11·81-s + 12·87-s − 6·89-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/3·9-s − 0.301·11-s − 1.10·13-s + 0.917·19-s + 1.25·23-s + 0.769·27-s − 1.11·29-s − 1.43·31-s + 0.348·33-s − 0.328·37-s + 1.28·39-s − 0.937·41-s − 1.21·43-s + 0.875·47-s + 0.824·53-s − 1.05·57-s + 1.56·59-s − 0.256·61-s + 1.22·67-s − 1.44·69-s − 1.42·71-s − 1.87·73-s + 0.900·79-s − 1.22·81-s + 1.28·87-s − 0.635·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 53900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(53900\)    =    \(2^{2} \cdot 5^{2} \cdot 7^{2} \cdot 11\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{53900} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 53900,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 \)
7 \( 1 \)
11 \( 1 + T \)
good3 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−14.72375255745627, −14.30740629235004, −13.53401353502352, −12.99541481829852, −12.70408070385397, −11.91717003827647, −11.68158372238616, −11.23007979369788, −10.61212185734258, −10.14810529517122, −9.695213884563099, −8.901392152599548, −8.643548857179000, −7.562647249067264, −7.284544826038956, −6.869224178403985, −6.051945045903717, −5.493481058423851, −5.119831353720627, −4.751667523696947, −3.765919339347261, −3.200646009539737, −2.421381298161214, −1.648049242869908, −0.7263066775360891, 0, 0.7263066775360891, 1.648049242869908, 2.421381298161214, 3.200646009539737, 3.765919339347261, 4.751667523696947, 5.119831353720627, 5.493481058423851, 6.051945045903717, 6.869224178403985, 7.284544826038956, 7.562647249067264, 8.643548857179000, 8.901392152599548, 9.695213884563099, 10.14810529517122, 10.61212185734258, 11.23007979369788, 11.68158372238616, 11.91717003827647, 12.70408070385397, 12.99541481829852, 13.53401353502352, 14.30740629235004, 14.72375255745627

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