Properties

Degree $4$
Conductor $9144576$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 7-s − 16·19-s + 7·25-s + 12·29-s + 10·31-s + 8·37-s + 12·47-s − 6·49-s + 6·53-s + 24·59-s + 30·83-s − 32·103-s + 40·109-s + 24·113-s − 5·121-s + 127-s + 131-s − 16·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 14·169-s + 173-s + ⋯
L(s)  = 1  + 0.377·7-s − 3.67·19-s + 7/5·25-s + 2.22·29-s + 1.79·31-s + 1.31·37-s + 1.75·47-s − 6/7·49-s + 0.824·53-s + 3.12·59-s + 3.29·83-s − 3.15·103-s + 3.83·109-s + 2.25·113-s − 0.454·121-s + 0.0887·127-s + 0.0873·131-s − 1.38·133-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 + 1.07·169-s + 0.0760·173-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

\(L(1)\) \(\approx\) \(3.141486335\)
\(L(\frac12)\) \(\approx\) \(3.141486335\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + p T^{2} \)
good5$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
71$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 146 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 119 T^{2} + p^{2} T^{4} \)
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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

−8.587022818580158515072550805783, −8.504142080063773725374715809751, −8.216205831991562722195419660954, −8.191060283550992492951725543615, −7.14797674551729065439939757771, −7.13181084832077600128675820844, −6.46948651493064809398827198121, −6.41970057766840032573831491298, −6.02015333512657292130556185901, −5.51879362875736544522655995494, −4.74066408934052204389732941642, −4.67910710826956394426360723021, −4.35741577887307563580969055375, −3.93725464834244646603198141873, −3.27849515130741377685159542255, −2.68860226482550610841304715849, −2.20761604454694210534814596204, −2.14621413959342333315168179823, −0.893600883325719009879333510516, −0.74172155904591400381856507566, 0.74172155904591400381856507566, 0.893600883325719009879333510516, 2.14621413959342333315168179823, 2.20761604454694210534814596204, 2.68860226482550610841304715849, 3.27849515130741377685159542255, 3.93725464834244646603198141873, 4.35741577887307563580969055375, 4.67910710826956394426360723021, 4.74066408934052204389732941642, 5.51879362875736544522655995494, 6.02015333512657292130556185901, 6.41970057766840032573831491298, 6.46948651493064809398827198121, 7.13181084832077600128675820844, 7.14797674551729065439939757771, 8.191060283550992492951725543615, 8.216205831991562722195419660954, 8.504142080063773725374715809751, 8.587022818580158515072550805783

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