Properties

Label 8-384e4-1.1-c1e4-0-3
Degree $8$
Conductor $21743271936$
Sign $1$
Analytic cond. $88.3961$
Root an. cond. $1.75107$
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

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{28} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(88.3961\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{384} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{28} \cdot 3^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.677455920\)
\(L(\frac12)\) \(\approx\) \(1.677455920\)
\(L(\frac{3}{2})\) 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

−8.187538888349644272478622506055, −7.908445510098098559161414722454, −7.71294832898232719335781053369, −7.54707509561433255840292960749, −7.22704519289053619322455979266, −7.13109257444879008780796679526, −6.71669739619605429367313171073, −6.43579796159215007122330409467, −6.13639680721720423643847145640, −5.94726273082563251809984699270, −5.59583465441700055415601694570, −5.43390876698687706175582951966, −5.35371331864920270843620974517, −4.61900307042811390647851823629, −4.53853832341200270929554884125, −4.05248489187420367544764072459, −4.00866200953003047777700149104, −3.79952816991674044677537430206, −3.29660434930338893539993961725, −2.97435160168075370410692631058, −2.41377703666323741835083524636, −2.02747850853675276182915285352, −1.96901064526864493807848384704, −1.27062072584849419492082993667, −0.52693761567553869632645893362, 0.52693761567553869632645893362, 1.27062072584849419492082993667, 1.96901064526864493807848384704, 2.02747850853675276182915285352, 2.41377703666323741835083524636, 2.97435160168075370410692631058, 3.29660434930338893539993961725, 3.79952816991674044677537430206, 4.00866200953003047777700149104, 4.05248489187420367544764072459, 4.53853832341200270929554884125, 4.61900307042811390647851823629, 5.35371331864920270843620974517, 5.43390876698687706175582951966, 5.59583465441700055415601694570, 5.94726273082563251809984699270, 6.13639680721720423643847145640, 6.43579796159215007122330409467, 6.71669739619605429367313171073, 7.13109257444879008780796679526, 7.22704519289053619322455979266, 7.54707509561433255840292960749, 7.71294832898232719335781053369, 7.908445510098098559161414722454, 8.187538888349644272478622506055

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