Properties

Degree $4$
Conductor $147456$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 9-s − 4·11-s − 8·13-s + 2·25-s − 4·27-s − 8·33-s − 8·37-s − 16·39-s − 16·47-s + 2·49-s − 4·59-s − 8·61-s + 16·71-s − 4·73-s + 4·75-s − 11·81-s + 20·83-s − 4·97-s − 4·99-s − 4·107-s − 8·109-s − 16·111-s − 8·117-s + 6·121-s + 127-s + 131-s + ⋯
L(s)  = 1  + 1.15·3-s + 1/3·9-s − 1.20·11-s − 2.21·13-s + 2/5·25-s − 0.769·27-s − 1.39·33-s − 1.31·37-s − 2.56·39-s − 2.33·47-s + 2/7·49-s − 0.520·59-s − 1.02·61-s + 1.89·71-s − 0.468·73-s + 0.461·75-s − 1.22·81-s + 2.19·83-s − 0.406·97-s − 0.402·99-s − 0.386·107-s − 0.766·109-s − 1.51·111-s − 0.739·117-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(147456\)    =    \(2^{14} \cdot 3^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{147456} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 147456,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
3$C_2$ \( 1 - 2 T + p T^{2} \)
good5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
13$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 66 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
59$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
61$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
71$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} ) \)
89$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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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

−9.265204394054927870253081026486, −8.415927036097658018462007127476, −8.080926924935224032946879214157, −7.77150526492255798109930537699, −7.18520237637046117380321878743, −6.85661286263459159950456749295, −6.07887016439203362608904379477, −5.22633510024610722329086500957, −5.04707257760722224952699405224, −4.45790123609111041692037088954, −3.49211587773070515732851046082, −3.03461745236516476789238084822, −2.42808900246114100784749020673, −1.90012581767396358727409595632, 0, 1.90012581767396358727409595632, 2.42808900246114100784749020673, 3.03461745236516476789238084822, 3.49211587773070515732851046082, 4.45790123609111041692037088954, 5.04707257760722224952699405224, 5.22633510024610722329086500957, 6.07887016439203362608904379477, 6.85661286263459159950456749295, 7.18520237637046117380321878743, 7.77150526492255798109930537699, 8.080926924935224032946879214157, 8.415927036097658018462007127476, 9.265204394054927870253081026486

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