Properties

Degree $4$
Conductor $23328$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 6·5-s − 8-s − 6·10-s − 8·13-s + 16-s + 6·20-s + 17·25-s + 8·26-s + 12·29-s − 32-s + 4·37-s − 6·40-s − 12·41-s − 13·49-s − 17·50-s − 8·52-s + 18·53-s − 12·58-s + 16·61-s + 64-s − 48·65-s − 14·73-s − 4·74-s + 6·80-s + 12·82-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 2.68·5-s − 0.353·8-s − 1.89·10-s − 2.21·13-s + 1/4·16-s + 1.34·20-s + 17/5·25-s + 1.56·26-s + 2.22·29-s − 0.176·32-s + 0.657·37-s − 0.948·40-s − 1.87·41-s − 1.85·49-s − 2.40·50-s − 1.10·52-s + 2.47·53-s − 1.57·58-s + 2.04·61-s + 1/8·64-s − 5.95·65-s − 1.63·73-s − 0.464·74-s + 0.670·80-s + 1.32·82-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23328 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23328 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(23328\)    =    \(2^{5} \cdot 3^{6}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{23328} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 23328,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.252252272\)
\(L(\frac12)\) \(\approx\) \(1.252252272\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
3 \( 1 \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 18 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
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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

−10.13919133269169682063036547080, −10.11526477822370291644410044895, −9.872045258839871807563899217385, −9.405159791372798554846122664326, −8.657482823158305382635077732253, −8.268048909391662018895990050852, −7.29929175763847648434390847974, −6.73860900674537002760989556093, −6.43760550649778711412630154056, −5.43287815412844995352588032205, −5.37363015537505503550836847886, −4.47356450228848094997188248522, −2.78595567768432254175608596610, −2.44850978333263328730546301919, −1.56286118008502235583804201463, 1.56286118008502235583804201463, 2.44850978333263328730546301919, 2.78595567768432254175608596610, 4.47356450228848094997188248522, 5.37363015537505503550836847886, 5.43287815412844995352588032205, 6.43760550649778711412630154056, 6.73860900674537002760989556093, 7.29929175763847648434390847974, 8.268048909391662018895990050852, 8.657482823158305382635077732253, 9.405159791372798554846122664326, 9.872045258839871807563899217385, 10.11526477822370291644410044895, 10.13919133269169682063036547080

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