Properties

Label 8-2e40-1.1-c1e4-0-6
Degree $8$
Conductor $1.100\times 10^{12}$
Sign $1$
Analytic cond. $4470.00$
Root an. cond. $2.85948$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 8·17-s + 32·47-s − 4·49-s + 32·79-s + 14·81-s − 8·97-s − 8·113-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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  + 1.94·17-s + 4.66·47-s − 4/7·49-s + 3.60·79-s + 14/9·81-s − 0.812·97-s − 0.752·113-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 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{40}\)
Sign: $1$
Analytic conductor: \(4470.00\)
Root analytic conductor: \(2.85948\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1024} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{40} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.147852940\)
\(L(\frac12)\) \(\approx\) \(4.147852940\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3$C_2^2$$\times$$C_2^2$ \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \)
5$C_2^2$$\times$$C_2^2$ \( ( 1 - 8 T^{2} + p^{2} T^{4} )( 1 + 8 T^{2} + p^{2} T^{4} ) \)
7$C_2^2$ \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^3$ \( 1 + 82 T^{4} + p^{4} T^{8} \)
13$C_2^3$ \( 1 + 146 T^{4} + p^{4} T^{8} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
19$C_2^2$$\times$$C_2^2$ \( ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} ) \)
23$C_2^2$ \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^3$ \( 1 - 1198 T^{4} + p^{4} T^{8} \)
31$C_2$ \( ( 1 + p T^{2} )^{4} \)
37$C_2^3$ \( 1 - 2062 T^{4} + p^{4} T^{8} \)
41$C_2^2$ \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^3$ \( 1 - 1198 T^{4} + p^{4} T^{8} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{4} \)
53$C_2^3$ \( 1 - 718 T^{4} + p^{4} T^{8} \)
59$C_2^3$ \( 1 - 878 T^{4} + p^{4} T^{8} \)
61$C_2^3$ \( 1 + 6482 T^{4} + p^{4} T^{8} \)
67$C_2^3$ \( 1 - 7822 T^{4} + p^{4} T^{8} \)
71$C_2^2$ \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{4} \)
83$C_2^3$ \( 1 + 3122 T^{4} + p^{4} T^{8} \)
89$C_2^2$ \( ( 1 - 174 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.22701457878079858275664800472, −7.17292089608055547753521233140, −6.51572067217407206485432866094, −6.48855825524144250605597903977, −6.21706140665644193498539726895, −5.88403925839702206332039992215, −5.75580977862261704534476117709, −5.71825374667437010823562266658, −5.13313904663649993546507091065, −5.07846918071791264040512175711, −5.04301086149437414324540959060, −4.60051403689155093043264581036, −4.13990520583006906587128290044, −4.04885733417651694282974305745, −3.90012383579256147622707480175, −3.52195504667435645060392012954, −3.22418325584603891664000756420, −3.05820101254712006132722471359, −2.72603965828437921003271189657, −2.29047467704905323043920468610, −2.08258155784625936442467934634, −1.84150029290889241218656329183, −1.10078576728659743524075207070, −0.949978495414823838712843208652, −0.57041881947983786115844130950, 0.57041881947983786115844130950, 0.949978495414823838712843208652, 1.10078576728659743524075207070, 1.84150029290889241218656329183, 2.08258155784625936442467934634, 2.29047467704905323043920468610, 2.72603965828437921003271189657, 3.05820101254712006132722471359, 3.22418325584603891664000756420, 3.52195504667435645060392012954, 3.90012383579256147622707480175, 4.04885733417651694282974305745, 4.13990520583006906587128290044, 4.60051403689155093043264581036, 5.04301086149437414324540959060, 5.07846918071791264040512175711, 5.13313904663649993546507091065, 5.71825374667437010823562266658, 5.75580977862261704534476117709, 5.88403925839702206332039992215, 6.21706140665644193498539726895, 6.48855825524144250605597903977, 6.51572067217407206485432866094, 7.17292089608055547753521233140, 7.22701457878079858275664800472

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