Properties

Label 8-768e4-1.1-c1e4-0-8
Degree $8$
Conductor $347892350976$
Sign $1$
Analytic cond. $1414.33$
Root an. cond. $2.47639$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·9-s + 20·25-s + 28·49-s − 8·73-s − 5·81-s − 40·97-s − 28·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 40·225-s + ⋯
L(s)  = 1  + 2/3·9-s + 4·25-s + 4·49-s − 0.936·73-s − 5/9·81-s − 4.06·97-s − 2.54·121-s + 0.0887·127-s + 0.0873·131-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 − 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 8/3·225-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{32} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(1414.33\)
Root analytic conductor: \(2.47639\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{32} \cdot 3^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.354911841\)
\(L(\frac12)\) \(\approx\) \(3.354911841\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
good5$C_2$ \( ( 1 - p T^{2} )^{4} \)
7$C_2$ \( ( 1 - p T^{2} )^{4} \)
11$C_2^2$ \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2$ \( ( 1 + p T^{2} )^{4} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \)
19$C_2^2$ \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{4} \)
29$C_2$ \( ( 1 - p T^{2} )^{4} \)
31$C_2$ \( ( 1 - p T^{2} )^{4} \)
37$C_2$ \( ( 1 + p T^{2} )^{4} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{4} \)
53$C_2$ \( ( 1 - p T^{2} )^{4} \)
59$C_2^2$ \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 + p T^{2} )^{4} \)
67$C_2^2$ \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{4} \)
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
79$C_2$ \( ( 1 - p T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + 158 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2$ \( ( 1 - 18 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.38401651784986402357208674302, −7.06271895056035254717550867783, −6.93060134003925553612437085584, −6.87910845864649643757293458251, −6.72970846236149838163561521362, −6.07423467668208618438010781175, −6.05071528493162728347776000837, −5.93994376959014222887536021304, −5.44908602401888469548068597467, −5.24882506383316024809690372661, −5.01467525339356990467809147284, −4.81480340417163280335683153532, −4.61348551058143328820193190301, −4.20664202398672353855469575432, −3.97080480939446642254851968417, −3.82977912038039599091042869748, −3.51304271182862339780075494675, −3.00644823746768916717868527813, −2.68191299890414399250769527500, −2.57633293308453698005659839446, −2.49621463928428395880358500249, −1.65445512759535782500617443695, −1.31184660084673937374665267419, −1.11773673340153941224648378812, −0.53428567744807779713219705668, 0.53428567744807779713219705668, 1.11773673340153941224648378812, 1.31184660084673937374665267419, 1.65445512759535782500617443695, 2.49621463928428395880358500249, 2.57633293308453698005659839446, 2.68191299890414399250769527500, 3.00644823746768916717868527813, 3.51304271182862339780075494675, 3.82977912038039599091042869748, 3.97080480939446642254851968417, 4.20664202398672353855469575432, 4.61348551058143328820193190301, 4.81480340417163280335683153532, 5.01467525339356990467809147284, 5.24882506383316024809690372661, 5.44908602401888469548068597467, 5.93994376959014222887536021304, 6.05071528493162728347776000837, 6.07423467668208618438010781175, 6.72970846236149838163561521362, 6.87910845864649643757293458251, 6.93060134003925553612437085584, 7.06271895056035254717550867783, 7.38401651784986402357208674302

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