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  − 2·5-s − 3·9-s + 2·11-s + 17-s − 2·19-s − 8·23-s − 25-s − 8·31-s + 4·37-s + 6·41-s − 4·43-s + 6·45-s − 8·47-s + 6·53-s − 4·55-s + 10·59-s + 10·61-s − 8·67-s + 4·71-s + 10·73-s − 4·79-s + 9·81-s − 6·83-s − 2·85-s + 6·89-s + 4·95-s + 14·97-s + ⋯
L(s)  = 1  − 0.894·5-s − 9-s + 0.603·11-s + 0.242·17-s − 0.458·19-s − 1.66·23-s − 1/5·25-s − 1.43·31-s + 0.657·37-s + 0.937·41-s − 0.609·43-s + 0.894·45-s − 1.16·47-s + 0.824·53-s − 0.539·55-s + 1.30·59-s + 1.28·61-s − 0.977·67-s + 0.474·71-s + 1.17·73-s − 0.450·79-s + 81-s − 0.658·83-s − 0.216·85-s + 0.635·89-s + 0.410·95-s + 1.42·97-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 \( 1 + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
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

−14.66559941031881, −14.32373250837898, −13.82366160707698, −13.05172172571249, −12.68696013858719, −11.95356054034812, −11.62156134499842, −11.33904658035005, −10.69903892675622, −9.992634830482615, −9.600277508236628, −8.731679060956944, −8.578003805151304, −7.797269696313333, −7.570759862108495, −6.737370724764039, −6.159135047641985, −5.702159888429872, −5.038299455131262, −4.244550258615855, −3.757253208071254, −3.371183077236614, −2.384355717048205, −1.881858896965633, −0.7390219862791393, 0, 0.7390219862791393, 1.881858896965633, 2.384355717048205, 3.371183077236614, 3.757253208071254, 4.244550258615855, 5.038299455131262, 5.702159888429872, 6.159135047641985, 6.737370724764039, 7.570759862108495, 7.797269696313333, 8.578003805151304, 8.731679060956944, 9.600277508236628, 9.992634830482615, 10.69903892675622, 11.33904658035005, 11.62156134499842, 11.95356054034812, 12.68696013858719, 13.05172172571249, 13.82366160707698, 14.32373250837898, 14.66559941031881

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