Properties

Degree $4$
Conductor $1408$
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-s − 2·4-s − 5-s − 2·7-s + 9-s − 5·11-s − 2·12-s + 2·13-s − 15-s + 4·16-s + 10·19-s + 2·20-s − 2·21-s − 5·23-s − 3·25-s + 4·27-s + 4·28-s + 2·29-s − 9·31-s − 5·33-s + 2·35-s − 2·36-s + 37-s + 2·39-s + 6·41-s − 4·43-s + 10·44-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s − 0.447·5-s − 0.755·7-s + 1/3·9-s − 1.50·11-s − 0.577·12-s + 0.554·13-s − 0.258·15-s + 16-s + 2.29·19-s + 0.447·20-s − 0.436·21-s − 1.04·23-s − 3/5·25-s + 0.769·27-s + 0.755·28-s + 0.371·29-s − 1.61·31-s − 0.870·33-s + 0.338·35-s − 1/3·36-s + 0.164·37-s + 0.320·39-s + 0.937·41-s − 0.609·43-s + 1.50·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1408 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1408 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1408\)    =    \(2^{7} \cdot 11\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{1408} (1, \cdot )$
Sato-Tate group: $\mathrm{USp}(4)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1408,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5194410224\)
\(L(\frac12)\) \(\approx\) \(0.5194410224\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T^{2} \)
11$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 6 T + p T^{2} ) \)
good3$C_2$$\times$$C_2$ \( ( 1 - p T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
7$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
23$D_{4}$ \( 1 + 5 T + 34 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
29$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$D_{4}$ \( 1 + 9 T + 46 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - T - p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 6 T + 58 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
61$D_{4}$ \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 3 T + 52 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - T + 34 T^{2} - p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 6 T + 82 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 19 T + 244 T^{2} + 19 p T^{3} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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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

−19.5024905413, −18.6515039551, −18.3241071710, −18.1547342285, −17.4256455054, −16.3143235701, −16.1014095962, −15.6947654200, −14.9477996999, −14.0965594736, −13.8527543598, −13.1622509030, −12.7575310708, −12.0782320852, −11.2323214140, −10.3430765805, −9.78142436710, −9.30631203129, −8.38167974442, −7.84889227658, −7.22430057203, −5.83270038767, −5.14016459947, −3.90550634524, −3.08349174391, 3.08349174391, 3.90550634524, 5.14016459947, 5.83270038767, 7.22430057203, 7.84889227658, 8.38167974442, 9.30631203129, 9.78142436710, 10.3430765805, 11.2323214140, 12.0782320852, 12.7575310708, 13.1622509030, 13.8527543598, 14.0965594736, 14.9477996999, 15.6947654200, 16.1014095962, 16.3143235701, 17.4256455054, 18.1547342285, 18.3241071710, 18.6515039551, 19.5024905413

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