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 − 2·7-s − 8-s + 2·14-s + 16-s − 12·23-s − 25-s − 2·28-s + 10·31-s − 32-s − 12·41-s + 12·46-s + 12·47-s − 11·49-s + 50-s + 2·56-s − 10·62-s + 64-s − 14·73-s + 16·79-s + 12·82-s − 36·89-s − 12·92-s − 12·94-s − 2·97-s + 11·98-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s + 0.534·14-s + 1/4·16-s − 2.50·23-s − 1/5·25-s − 0.377·28-s + 1.79·31-s − 0.176·32-s − 1.87·41-s + 1.76·46-s + 1.75·47-s − 1.57·49-s + 0.141·50-s + 0.267·56-s − 1.27·62-s + 1/8·64-s − 1.63·73-s + 1.80·79-s + 1.32·82-s − 3.81·89-s − 1.25·92-s − 1.23·94-s − 0.203·97-s + 1.11·98-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$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
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

−9.572383575127331479120897865207, −8.853339056894194014756168928413, −8.268048909391662018895990050852, −8.104088285674841823256361524017, −7.46895104625204893367477999066, −6.73860900674537002760989556093, −6.47796475592373752838072674203, −5.89467970712550736408970851242, −5.37363015537505503550836847886, −4.43365380485938957073214377677, −3.92407328889816322387260499130, −3.12927788056823217499419163319, −2.44850978333263328730546301919, −1.52887349549650490954756981970, 0, 1.52887349549650490954756981970, 2.44850978333263328730546301919, 3.12927788056823217499419163319, 3.92407328889816322387260499130, 4.43365380485938957073214377677, 5.37363015537505503550836847886, 5.89467970712550736408970851242, 6.47796475592373752838072674203, 6.73860900674537002760989556093, 7.46895104625204893367477999066, 8.104088285674841823256361524017, 8.268048909391662018895990050852, 8.853339056894194014756168928413, 9.572383575127331479120897865207

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