Properties

Degree $2$
Conductor $53312$
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 − 6·11-s + 2·13-s + 17-s − 4·19-s − 5·25-s + 4·27-s + 4·31-s + 12·33-s + 4·37-s − 4·39-s − 6·41-s − 8·43-s − 2·51-s + 6·53-s + 8·57-s − 4·61-s − 8·67-s − 2·73-s + 10·75-s + 8·79-s − 11·81-s + 6·89-s − 8·93-s − 14·97-s − 6·99-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/3·9-s − 1.80·11-s + 0.554·13-s + 0.242·17-s − 0.917·19-s − 25-s + 0.769·27-s + 0.718·31-s + 2.08·33-s + 0.657·37-s − 0.640·39-s − 0.937·41-s − 1.21·43-s − 0.280·51-s + 0.824·53-s + 1.05·57-s − 0.512·61-s − 0.977·67-s − 0.234·73-s + 1.15·75-s + 0.900·79-s − 1.22·81-s + 0.635·89-s − 0.829·93-s − 1.42·97-s − 0.603·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(53312\)    =    \(2^{6} \cdot 7^{2} \cdot 17\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{53312} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 53312,\ (\ :1/2),\ -1)\)

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

−14.86388560302725, −14.09337028347429, −13.45073880483394, −13.22440755714373, −12.63661125574016, −12.07395734312455, −11.59144138148356, −11.16196691972959, −10.53683020606456, −10.26320301257389, −9.838645102058224, −8.852348694823677, −8.417866511611559, −7.867283413873452, −7.363734512408603, −6.571340862687066, −6.082386861388870, −5.716569584399655, −4.977997234925646, −4.746851003745291, −3.859708650057943, −3.114301084678027, −2.442987704118711, −1.702342925604259, −0.6625528198484499, 0, 0.6625528198484499, 1.702342925604259, 2.442987704118711, 3.114301084678027, 3.859708650057943, 4.746851003745291, 4.977997234925646, 5.716569584399655, 6.082386861388870, 6.571340862687066, 7.363734512408603, 7.867283413873452, 8.417866511611559, 8.852348694823677, 9.838645102058224, 10.26320301257389, 10.53683020606456, 11.16196691972959, 11.59144138148356, 12.07395734312455, 12.63661125574016, 13.22440755714373, 13.45073880483394, 14.09337028347429, 14.86388560302725

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