Properties

Degree $4$
Conductor $688$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 2·5-s + 2·9-s − 3·11-s + 13-s + 4·15-s + 3·17-s + 2·19-s + 5·23-s + 2·25-s − 6·27-s − 4·29-s + 31-s + 6·33-s + 2·37-s − 2·39-s − 13·41-s + 5·43-s − 4·45-s − 10·49-s − 6·51-s + 53-s + 6·55-s − 4·57-s − 4·59-s + 4·61-s − 2·65-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.894·5-s + 2/3·9-s − 0.904·11-s + 0.277·13-s + 1.03·15-s + 0.727·17-s + 0.458·19-s + 1.04·23-s + 2/5·25-s − 1.15·27-s − 0.742·29-s + 0.179·31-s + 1.04·33-s + 0.328·37-s − 0.320·39-s − 2.03·41-s + 0.762·43-s − 0.596·45-s − 1.42·49-s − 0.840·51-s + 0.137·53-s + 0.809·55-s − 0.529·57-s − 0.520·59-s + 0.512·61-s − 0.248·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(688\)    =    \(2^{4} \cdot 43\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{688} (1, \cdot )$
Sato-Tate group: $\mathrm{USp}(4)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 688,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3213412268\)
\(L(\frac12)\) \(\approx\) \(0.3213412268\)
\(L(\frac{3}{2})\) 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 \( 1 \)
43$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 4 T + p T^{2} ) \)
good3$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$D_{4}$ \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - T + 4 T^{2} - p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 5 T + 30 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - T + 30 T^{2} - p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 2 T - 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 13 T + 92 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - T + 44 T^{2} - p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 4 T - 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 4 T + 86 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 3 T + 34 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 10 T + 78 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 8 T + 142 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 5 T - 30 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−19.3642777917, −18.8244228195, −18.4739098555, −17.8018846229, −17.0729077835, −16.6871462113, −16.0755100110, −15.4407543475, −15.0821250386, −14.1261448667, −13.2462848409, −12.7447459431, −11.9944835953, −11.4031431055, −10.9836553937, −10.1883900718, −9.37318708003, −8.20890805774, −7.58771620437, −6.70484591592, −5.59619953870, −4.91804156769, −3.50521274292, 3.50521274292, 4.91804156769, 5.59619953870, 6.70484591592, 7.58771620437, 8.20890805774, 9.37318708003, 10.1883900718, 10.9836553937, 11.4031431055, 11.9944835953, 12.7447459431, 13.2462848409, 14.1261448667, 15.0821250386, 15.4407543475, 16.0755100110, 16.6871462113, 17.0729077835, 17.8018846229, 18.4739098555, 18.8244228195, 19.3642777917

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