Properties

Degree $4$
Conductor $9144576$
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  + 5·7-s − 2·19-s + 10·25-s + 8·31-s + 2·37-s + 18·49-s + 14·103-s + 4·109-s + 22·121-s + 127-s + 131-s − 10·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 23·169-s + 173-s + 50·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  + 1.88·7-s − 0.458·19-s + 2·25-s + 1.43·31-s + 0.328·37-s + 18/7·49-s + 1.37·103-s + 0.383·109-s + 2·121-s + 0.0887·127-s + 0.0873·131-s − 0.867·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.76·169-s + 0.0760·173-s + 3.77·175-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9144576\)    =    \(2^{8} \cdot 3^{6} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{3024} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9144576,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.213234835\)
\(L(\frac12)\) \(\approx\) \(4.213234835\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 - 5 T + p T^{2} \)
good5$C_2$ \( ( 1 - p T^{2} )^{2} \)
11$C_2$ \( ( 1 - p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
71$C_2$ \( ( 1 - p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
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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

−8.666704125815051460347817594089, −8.578426893562934369830618639324, −8.191198838437173175667719206220, −7.911554168247869762044599229253, −7.34151135788103775123796448684, −7.22706658154691271852652996124, −6.55611009572176393682170974254, −6.41651793704641959547709427338, −5.74309944355885487834802716971, −5.45569059392123083395053552887, −4.83376066808061143982723579449, −4.79353230927762537161910130736, −4.31466223597804429417569000636, −3.98672837724002051336518278271, −3.10106985226039559831275240406, −2.95120222167868351862903863597, −2.15677355281794249529549971710, −1.89869309425830551682810761125, −1.11333300124689030659159146722, −0.75589049633196307532403972979, 0.75589049633196307532403972979, 1.11333300124689030659159146722, 1.89869309425830551682810761125, 2.15677355281794249529549971710, 2.95120222167868351862903863597, 3.10106985226039559831275240406, 3.98672837724002051336518278271, 4.31466223597804429417569000636, 4.79353230927762537161910130736, 4.83376066808061143982723579449, 5.45569059392123083395053552887, 5.74309944355885487834802716971, 6.41651793704641959547709427338, 6.55611009572176393682170974254, 7.22706658154691271852652996124, 7.34151135788103775123796448684, 7.911554168247869762044599229253, 8.191198838437173175667719206220, 8.578426893562934369830618639324, 8.666704125815051460347817594089

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