Properties

Degree $4$
Conductor $93312$
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 + 16-s + 4·19-s + 6·20-s − 12·23-s + 17·25-s + 12·29-s − 32-s − 4·38-s − 6·40-s − 20·43-s + 12·46-s + 12·47-s − 13·49-s − 17·50-s + 18·53-s − 12·58-s + 64-s + 28·67-s − 14·73-s + 4·76-s + 6·80-s + 20·86-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 2.68·5-s − 0.353·8-s − 1.89·10-s + 1/4·16-s + 0.917·19-s + 1.34·20-s − 2.50·23-s + 17/5·25-s + 2.22·29-s − 0.176·32-s − 0.648·38-s − 0.948·40-s − 3.04·43-s + 1.76·46-s + 1.75·47-s − 1.85·49-s − 2.40·50-s + 2.47·53-s − 1.57·58-s + 1/8·64-s + 3.42·67-s − 1.63·73-s + 0.458·76-s + 0.670·80-s + 2.15·86-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93312 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93312 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(93312\)    =    \(2^{7} \cdot 3^{6}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{93312} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 93312,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.808484862\)
\(L(\frac12)\) \(\approx\) \(1.808484862\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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

−9.872045258839871807563899217385, −9.300538826904107531692188360431, −8.699511810288277483701861071116, −8.268048909391662018895990050852, −7.82087500680797002813709866504, −6.73860900674537002760989556093, −6.68641118928175950907657181471, −6.10265864740654543835243104495, −5.43217621063437274714591238543, −5.37363015537505503550836847886, −4.34994968190190118318545746782, −3.35914372705473359723249826168, −2.44850978333263328730546301919, −2.05899421145709367031472603742, −1.26980488075185704480426069120, 1.26980488075185704480426069120, 2.05899421145709367031472603742, 2.44850978333263328730546301919, 3.35914372705473359723249826168, 4.34994968190190118318545746782, 5.37363015537505503550836847886, 5.43217621063437274714591238543, 6.10265864740654543835243104495, 6.68641118928175950907657181471, 6.73860900674537002760989556093, 7.82087500680797002813709866504, 8.268048909391662018895990050852, 8.699511810288277483701861071116, 9.300538826904107531692188360431, 9.872045258839871807563899217385

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