Properties

Degree $4$
Conductor $93312$
Sign $-1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 8-s − 6·11-s + 16-s + 4·19-s + 6·22-s − 25-s − 32-s − 4·38-s − 12·41-s − 20·43-s − 6·44-s − 13·49-s + 50-s + 24·59-s + 64-s + 28·67-s − 14·73-s + 4·76-s + 12·82-s − 6·83-s + 20·86-s + 6·88-s − 36·89-s − 2·97-s + 13·98-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s − 1.80·11-s + 1/4·16-s + 0.917·19-s + 1.27·22-s − 1/5·25-s − 0.176·32-s − 0.648·38-s − 1.87·41-s − 3.04·43-s − 0.904·44-s − 1.85·49-s + 0.141·50-s + 3.12·59-s + 1/8·64-s + 3.42·67-s − 1.63·73-s + 0.458·76-s + 1.32·82-s − 0.658·83-s + 2.15·86-s + 0.639·88-s − 3.81·89-s − 0.203·97-s + 1.31·98-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: \(1\)
Selberg data: \((4,\ 93312,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 + T \)
3 \( 1 \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
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$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
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

−9.689943470574670322603802395819, −8.646897306767688973776858314960, −8.268048909391662018895990050852, −8.177655093885067516195802420137, −7.44444168021687832363479010767, −6.73860900674537002760989556093, −6.72818725632924944067815958116, −5.48544965844839209036398212504, −5.37363015537505503550836847886, −4.81849202265609017039962796294, −3.72379428045727412120464814901, −3.13977056925804084341234249361, −2.44850978333263328730546301919, −1.55100613792804333036729509228, 0, 1.55100613792804333036729509228, 2.44850978333263328730546301919, 3.13977056925804084341234249361, 3.72379428045727412120464814901, 4.81849202265609017039962796294, 5.37363015537505503550836847886, 5.48544965844839209036398212504, 6.72818725632924944067815958116, 6.73860900674537002760989556093, 7.44444168021687832363479010767, 8.177655093885067516195802420137, 8.268048909391662018895990050852, 8.646897306767688973776858314960, 9.689943470574670322603802395819

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