Properties

Label 8-320e4-1.1-c1e4-0-2
Degree $8$
Conductor $10485760000$
Sign $1$
Analytic cond. $42.6293$
Root an. cond. $1.59850$
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  + 10·25-s − 24·29-s + 2·81-s − 24·89-s + 72·101-s − 44·121-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 + 227-s + 229-s + ⋯
L(s)  = 1  + 2·25-s − 4.45·29-s + 2/9·81-s − 2.54·89-s + 7.16·101-s − 4·121-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 + 0.0663·227-s + 0.0660·229-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(42.6293\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{320} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 5^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.457780710\)
\(L(\frac12)\) \(\approx\) \(1.457780710\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( ( 1 - p T^{2} )^{2} \)
good3$C_2^2$$\times$$C_2^2$ \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} ) \)
7$C_2^2$$\times$$C_2^2$ \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \)
11$C_2$ \( ( 1 + p T^{2} )^{4} \)
13$C_2$ \( ( 1 - p T^{2} )^{4} \)
17$C_2$ \( ( 1 - p T^{2} )^{4} \)
19$C_2$ \( ( 1 + p T^{2} )^{4} \)
23$C_2^2$$\times$$C_2^2$ \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} ) \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{4} \)
37$C_2$ \( ( 1 - p T^{2} )^{4} \)
41$C_2^2$ \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$$\times$$C_2^2$ \( ( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} ) \)
47$C_2^2$$\times$$C_2^2$ \( ( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} )( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} ) \)
53$C_2$ \( ( 1 - p T^{2} )^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{4} \)
61$C_2^2$ \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2^2$$\times$$C_2^2$ \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{4} \)
73$C_2$ \( ( 1 - p T^{2} )^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{4} \)
83$C_2^2$$\times$$C_2^2$ \( ( 1 - 22 T + 242 T^{2} - 22 p T^{3} + p^{2} T^{4} )( 1 + 22 T + 242 T^{2} + 22 p T^{3} + p^{2} T^{4} ) \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
97$C_2$ \( ( 1 - p T^{2} )^{4} \)
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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

−8.595704134806719994888197460077, −8.145333263117380176796508711167, −7.88735130799031441078529146259, −7.59929071574645992488737823723, −7.55268921339908238736794427242, −7.22137510876829682312366512447, −6.93029773178061746386814311319, −6.70028884169748980067068633430, −6.50484342918042029504547676519, −5.93781900484777146602802890623, −5.88291129271952006489432174005, −5.57001803235916423991812242157, −5.37405656422761824414783060441, −4.86724775767359282115239252340, −4.83253584996688005409165278981, −4.41863975025615865808637163119, −3.96393813181188993381959928447, −3.68197503671921374509558661383, −3.52816577984308970527095517834, −3.07259212614449354672858705353, −2.70872896647222072483784235026, −2.21309292232618562980695288057, −1.82479599878298816971131228526, −1.44690429251901538790744228020, −0.54203674593918975402133761281, 0.54203674593918975402133761281, 1.44690429251901538790744228020, 1.82479599878298816971131228526, 2.21309292232618562980695288057, 2.70872896647222072483784235026, 3.07259212614449354672858705353, 3.52816577984308970527095517834, 3.68197503671921374509558661383, 3.96393813181188993381959928447, 4.41863975025615865808637163119, 4.83253584996688005409165278981, 4.86724775767359282115239252340, 5.37405656422761824414783060441, 5.57001803235916423991812242157, 5.88291129271952006489432174005, 5.93781900484777146602802890623, 6.50484342918042029504547676519, 6.70028884169748980067068633430, 6.93029773178061746386814311319, 7.22137510876829682312366512447, 7.55268921339908238736794427242, 7.59929071574645992488737823723, 7.88735130799031441078529146259, 8.145333263117380176796508711167, 8.595704134806719994888197460077

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