Properties

Degree $4$
Conductor $5364$
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-s − 4-s − 2·5-s − 6·7-s + 3·8-s + 9-s + 2·10-s − 2·11-s − 13-s + 6·14-s − 16-s − 18-s − 8·19-s + 2·20-s + 2·22-s + 2·23-s + 25-s + 26-s + 6·28-s + 29-s − 2·31-s − 5·32-s + 12·35-s − 36-s + 2·37-s + 8·38-s − 6·40-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 0.894·5-s − 2.26·7-s + 1.06·8-s + 1/3·9-s + 0.632·10-s − 0.603·11-s − 0.277·13-s + 1.60·14-s − 1/4·16-s − 0.235·18-s − 1.83·19-s + 0.447·20-s + 0.426·22-s + 0.417·23-s + 1/5·25-s + 0.196·26-s + 1.13·28-s + 0.185·29-s − 0.359·31-s − 0.883·32-s + 2.02·35-s − 1/6·36-s + 0.328·37-s + 1.29·38-s − 0.948·40-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5364 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5364 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5364\)    =    \(2^{2} \cdot 3^{2} \cdot 149\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{5364} (1, \cdot )$
Sato-Tate group: $\mathrm{USp}(4)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 5364,\ (\ :1/2, 1/2),\ -1)\)

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$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( 1 + T + p T^{2} \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
149$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 15 T + p T^{2} ) \)
good5$D_{4}$ \( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 6 T + 3 p T^{2} + 6 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 8 T + 35 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 2 T + 24 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - T + 32 T^{2} - p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
41$D_{4}$ \( 1 - 3 T + 40 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 2 T + 20 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 6 T + 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 10 T + 62 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
61$C_4$ \( 1 + T - 24 T^{2} + p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
73$D_{4}$ \( 1 - 2 T - 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 4 T + 26 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 4 T + 28 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 9 T + 74 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.7584272782, −16.9075420170, −16.7370039493, −16.2409470601, −15.7688144620, −15.2566696675, −14.8216314321, −13.9479046739, −13.2269439962, −13.0273447754, −12.5726778172, −12.0579192258, −11.0740332976, −10.4391455915, −10.1955837979, −9.42648785635, −9.04972485077, −8.35472402438, −7.68940970807, −6.99691240167, −6.47764556293, −5.53429194409, −4.36634971385, −3.83388295612, −2.74752210577, 0, 2.74752210577, 3.83388295612, 4.36634971385, 5.53429194409, 6.47764556293, 6.99691240167, 7.68940970807, 8.35472402438, 9.04972485077, 9.42648785635, 10.1955837979, 10.4391455915, 11.0740332976, 12.0579192258, 12.5726778172, 13.0273447754, 13.2269439962, 13.9479046739, 14.8216314321, 15.2566696675, 15.7688144620, 16.2409470601, 16.7370039493, 16.9075420170, 17.7584272782

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