Properties

Degree $4$
Conductor $278784$
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  + 2·3-s − 4·5-s + 3·9-s − 4·11-s − 8·15-s + 16·23-s + 2·25-s + 4·27-s − 16·31-s − 8·33-s + 12·37-s − 12·45-s − 14·49-s − 4·53-s + 16·55-s − 8·59-s + 8·67-s + 32·69-s − 16·71-s + 4·75-s + 5·81-s − 12·89-s − 32·93-s + 4·97-s − 12·99-s − 32·103-s + 24·111-s + ⋯
L(s)  = 1  + 1.15·3-s − 1.78·5-s + 9-s − 1.20·11-s − 2.06·15-s + 3.33·23-s + 2/5·25-s + 0.769·27-s − 2.87·31-s − 1.39·33-s + 1.97·37-s − 1.78·45-s − 2·49-s − 0.549·53-s + 2.15·55-s − 1.04·59-s + 0.977·67-s + 3.85·69-s − 1.89·71-s + 0.461·75-s + 5/9·81-s − 1.27·89-s − 3.31·93-s + 0.406·97-s − 1.20·99-s − 3.15·103-s + 2.27·111-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 278784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 278784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(278784\)    =    \(2^{8} \cdot 3^{2} \cdot 11^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{278784} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 278784,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( ( 1 - T )^{2} \)
11$C_2$ \( 1 + 4 T + p T^{2} \)
good5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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

−8.541652558871803274196386006614, −8.173282349834764667604075089133, −7.63618059284105349520809602510, −7.32438284862293693328503541547, −7.26958585488936859387169650645, −6.42508206754472676439323141059, −5.62755823403878952826962811199, −5.01357758335939619070346679818, −4.60156021497859087443489366122, −4.01628213317951587527532774984, −3.32790226181780202269440904381, −3.14826068230388029805668446658, −2.43505247067401554863084291555, −1.37409591703931866594987960910, 0, 1.37409591703931866594987960910, 2.43505247067401554863084291555, 3.14826068230388029805668446658, 3.32790226181780202269440904381, 4.01628213317951587527532774984, 4.60156021497859087443489366122, 5.01357758335939619070346679818, 5.62755823403878952826962811199, 6.42508206754472676439323141059, 7.26958585488936859387169650645, 7.32438284862293693328503541547, 7.63618059284105349520809602510, 8.173282349834764667604075089133, 8.541652558871803274196386006614

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