Properties

Degree $2$
Conductor $53312$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 2·5-s + 6·9-s + 5·11-s + 7·13-s + 6·15-s − 17-s + 2·19-s − 25-s − 9·27-s − 6·29-s + 4·31-s − 15·33-s + 8·37-s − 21·39-s + 2·41-s + 8·43-s − 12·45-s − 10·47-s + 3·51-s + 3·53-s − 10·55-s − 6·57-s − 12·61-s − 14·65-s + 2·67-s − 71-s + ⋯
L(s)  = 1  − 1.73·3-s − 0.894·5-s + 2·9-s + 1.50·11-s + 1.94·13-s + 1.54·15-s − 0.242·17-s + 0.458·19-s − 1/5·25-s − 1.73·27-s − 1.11·29-s + 0.718·31-s − 2.61·33-s + 1.31·37-s − 3.36·39-s + 0.312·41-s + 1.21·43-s − 1.78·45-s − 1.45·47-s + 0.420·51-s + 0.412·53-s − 1.34·55-s − 0.794·57-s − 1.53·61-s − 1.73·65-s + 0.244·67-s − 0.118·71-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: \(0\)
Selberg data: \((2,\ 53312,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.146317487\)
\(L(\frac12)\) \(\approx\) \(1.146317487\)
\(L(\frac{3}{2})\) 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 \( 1 + p T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 - 7 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
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 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−14.52302527754722, −13.83980860034422, −13.28173239879062, −12.80974630849655, −12.19285541378714, −11.68407657194448, −11.46619402404985, −11.00159868955627, −10.73030889333012, −9.770032551632455, −9.424140621796663, −8.703741667261129, −8.124099181997970, −7.466230780190200, −6.885755664443868, −6.379334569687012, −5.906035185035232, −5.601527727791619, −4.544755224516389, −4.195544060627674, −3.808607094170241, −3.029294025145286, −1.614990594337569, −1.202075266593297, −0.4908803065005152, 0.4908803065005152, 1.202075266593297, 1.614990594337569, 3.029294025145286, 3.808607094170241, 4.195544060627674, 4.544755224516389, 5.601527727791619, 5.906035185035232, 6.379334569687012, 6.885755664443868, 7.466230780190200, 8.124099181997970, 8.703741667261129, 9.424140621796663, 9.770032551632455, 10.73030889333012, 11.00159868955627, 11.46619402404985, 11.68407657194448, 12.19285541378714, 12.80974630849655, 13.28173239879062, 13.83980860034422, 14.52302527754722

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