Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·9-s − 5·11-s + 5·13-s − 17-s − 6·19-s − 4·23-s − 5·25-s + 5·27-s − 4·29-s + 5·33-s − 8·37-s − 5·39-s + 4·41-s − 6·43-s + 6·47-s + 51-s − 11·53-s + 6·57-s + 10·59-s + 10·67-s + 4·69-s − 9·71-s + 4·73-s + 5·75-s − 9·79-s + 81-s + ⋯
L(s)  = 1  − 0.577·3-s − 2/3·9-s − 1.50·11-s + 1.38·13-s − 0.242·17-s − 1.37·19-s − 0.834·23-s − 25-s + 0.962·27-s − 0.742·29-s + 0.870·33-s − 1.31·37-s − 0.800·39-s + 0.624·41-s − 0.914·43-s + 0.875·47-s + 0.140·51-s − 1.51·53-s + 0.794·57-s + 1.30·59-s + 1.22·67-s + 0.481·69-s − 1.06·71-s + 0.468·73-s + 0.577·75-s − 1.01·79-s + 1/9·81-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: \(2\)
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 \( 1 + T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 9 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−15.05717328147490, −14.42276262649320, −13.71916759037734, −13.56394695939214, −12.78136711795994, −12.57194176501754, −11.80205861140399, −11.18391057639351, −11.01639998247252, −10.39232342856532, −10.03427912980581, −9.192257451737906, −8.556860173216121, −8.237254550473836, −7.780419518967847, −6.909452695721244, −6.401317976463480, −5.758701240317544, −5.547650915874713, −4.838262658949160, −4.039869369399232, −3.604116378661595, −2.698544686448879, −2.163681771070643, −1.355447021893490, 0, 0, 1.355447021893490, 2.163681771070643, 2.698544686448879, 3.604116378661595, 4.039869369399232, 4.838262658949160, 5.547650915874713, 5.758701240317544, 6.401317976463480, 6.909452695721244, 7.780419518967847, 8.237254550473836, 8.556860173216121, 9.192257451737906, 10.03427912980581, 10.39232342856532, 11.01639998247252, 11.18391057639351, 11.80205861140399, 12.57194176501754, 12.78136711795994, 13.56394695939214, 13.71916759037734, 14.42276262649320, 15.05717328147490

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