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  + 7-s + 14·19-s + 10·25-s − 8·31-s − 22·37-s − 6·49-s − 26·103-s + 4·109-s + 22·121-s + 127-s + 131-s + 14·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + 10·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  + 0.377·7-s + 3.21·19-s + 2·25-s − 1.43·31-s − 3.61·37-s − 6/7·49-s − 2.56·103-s + 0.383·109-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 1.21·133-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 − 0.0769·169-s + 0.0760·173-s + 0.755·175-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

\(L(1)\) \(\approx\) \(2.654171628\)
\(L(\frac12)\) \(\approx\) \(2.654171628\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - p T^{2} )^{2} \)
11$C_2$ \( ( 1 - p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
17$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
67$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
71$C_2$ \( ( 1 - p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 19 T + p T^{2} )( 1 + 19 T + p T^{2} ) \)
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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

−8.906963346491445421921039671331, −8.421894219359877368881252280662, −8.348423686020990509799095446587, −7.71785976537577710025475802949, −7.17612776508977148785431657613, −7.17005599180642840661946160726, −6.91927548056159077734480124059, −6.28782117467784757962816402890, −5.67859675803281510776382515499, −5.26563074865198188127090752944, −5.25961503727527862990879467740, −4.81260312517834832111990639279, −4.26688041801989086698467950158, −3.50960176488685883175680390551, −3.29950634425457247007850170390, −3.10872546596083367057783420229, −2.32695934934257295661406682749, −1.57283632837342061320247865016, −1.36466139652577772071054600575, −0.53776549503771593449730257695, 0.53776549503771593449730257695, 1.36466139652577772071054600575, 1.57283632837342061320247865016, 2.32695934934257295661406682749, 3.10872546596083367057783420229, 3.29950634425457247007850170390, 3.50960176488685883175680390551, 4.26688041801989086698467950158, 4.81260312517834832111990639279, 5.25961503727527862990879467740, 5.26563074865198188127090752944, 5.67859675803281510776382515499, 6.28782117467784757962816402890, 6.91927548056159077734480124059, 7.17005599180642840661946160726, 7.17612776508977148785431657613, 7.71785976537577710025475802949, 8.348423686020990509799095446587, 8.421894219359877368881252280662, 8.906963346491445421921039671331

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