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

\(L(1)\) \(\approx\) \(1.404411611\)
\(L(\frac12)\) \(\approx\) \(1.404411611\)
\(L(\frac{3}{2})\) 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

−8.940862497842335180257010594715, −8.730946668540202363779368830964, −8.196928458065173802739480179206, −7.68864693948181262542318317339, −7.31236684886691527185828898520, −7.00087654990579005132539029906, −6.63235754814692409934266158721, −6.40583952799669887976218667675, −5.85617729892471095589469647032, −5.54080668261508826367836218049, −5.15488220060993446885474017736, −4.63284382844607820728718813521, −4.11460996600220646220158461440, −3.71530613657385921574732772242, −3.19200314612979602348259899630, −2.99707027982598922428588140467, −2.50104724660868707014546709584, −1.83324001589955420355968465519, −1.04762214568876821653473194168, −0.42674313281574924849884234814, 0.42674313281574924849884234814, 1.04762214568876821653473194168, 1.83324001589955420355968465519, 2.50104724660868707014546709584, 2.99707027982598922428588140467, 3.19200314612979602348259899630, 3.71530613657385921574732772242, 4.11460996600220646220158461440, 4.63284382844607820728718813521, 5.15488220060993446885474017736, 5.54080668261508826367836218049, 5.85617729892471095589469647032, 6.40583952799669887976218667675, 6.63235754814692409934266158721, 7.00087654990579005132539029906, 7.31236684886691527185828898520, 7.68864693948181262542318317339, 8.196928458065173802739480179206, 8.730946668540202363779368830964, 8.940862497842335180257010594715

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