Properties

Label 8-1400e4-1.1-c0e4-0-1
Degree $8$
Conductor $3.842\times 10^{12}$
Sign $1$
Analytic cond. $0.238309$
Root an. cond. $0.835877$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 2·9-s + 6·31-s − 2·36-s + 49-s − 64-s + 4·71-s + 2·79-s + 81-s − 6·89-s − 2·121-s + 6·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 4-s − 2·9-s + 6·31-s − 2·36-s + 49-s − 64-s + 4·71-s + 2·79-s + 81-s − 6·89-s − 2·121-s + 6·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 5^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(0.238309\)
Root analytic conductor: \(0.835877\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1400} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.223311715\)
\(L(\frac12)\) \(\approx\) \(1.223311715\)
\(L(1)\) 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$C_2^2$ \( 1 - T^{2} + T^{4} \)
5 \( 1 \)
7$C_2^2$ \( 1 - T^{2} + T^{4} \)
good3$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
11$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
17$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
19$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
23$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
31$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 - T + T^{2} )^{2} \)
37$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
41$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
43$C_2$ \( ( 1 + T^{2} )^{4} \)
47$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
53$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
71$C_2$ \( ( 1 - T + T^{2} )^{4} \)
73$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
79$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
89$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 + T + T^{2} )^{2} \)
97$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
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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

−6.86947400123642982821511951848, −6.69196782425426827306123981798, −6.63532427232161056470101490968, −6.55040576083291127361691162647, −6.17247791406425204569962969618, −5.95474826851794784163257743483, −5.72505500947506354828497585868, −5.67685385553752730178630997665, −5.34142690993631580743450149420, −5.02172893201302514963705171709, −4.88010053989908111659696060361, −4.56140998207264830748760057485, −4.34113853677316512842851802228, −4.25678018355909960867551659412, −3.74530240559608923087225185491, −3.47160226851284291616910371761, −3.35286231215934185708487548345, −2.81208054691695784311617190260, −2.69880606878090054973636048477, −2.54372587052313966033813673154, −2.45628824756722494273533155425, −2.17577986626440423989841620957, −1.31612938331364370024605107047, −1.30472567027342386452105907802, −0.74138255448435302885048051543, 0.74138255448435302885048051543, 1.30472567027342386452105907802, 1.31612938331364370024605107047, 2.17577986626440423989841620957, 2.45628824756722494273533155425, 2.54372587052313966033813673154, 2.69880606878090054973636048477, 2.81208054691695784311617190260, 3.35286231215934185708487548345, 3.47160226851284291616910371761, 3.74530240559608923087225185491, 4.25678018355909960867551659412, 4.34113853677316512842851802228, 4.56140998207264830748760057485, 4.88010053989908111659696060361, 5.02172893201302514963705171709, 5.34142690993631580743450149420, 5.67685385553752730178630997665, 5.72505500947506354828497585868, 5.95474826851794784163257743483, 6.17247791406425204569962969618, 6.55040576083291127361691162647, 6.63532427232161056470101490968, 6.69196782425426827306123981798, 6.86947400123642982821511951848

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