↑

Properties

Label 8-2016e4-1.1-c0e4-0-15
Degree $8$
Conductor $1.652\times 10^{13}$
Sign $1$
Analytic cond. $1.02468$
Root an. cond. $1.00305$
Motivic weight $0$
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  + 4·2-s + 10·4-s + 20·8-s + 35·16-s + 56·32-s − 4·37-s + 84·64-s − 4·67-s − 4·71-s − 16·74-s − 8·113-s − 2·121-s + 127-s + 120·128-s + 131-s − 16·134-s + 137-s + 139-s − 16·142-s − 40·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + ⋯
L(s)  = 1  + 4·2-s + 10·4-s + 20·8-s + 35·16-s + 56·32-s − 4·37-s + 84·64-s − 4·67-s − 4·71-s − 16·74-s − 8·113-s − 2·121-s + 127-s + 120·128-s + 131-s − 16·134-s + 137-s + 139-s − 16·142-s − 40·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{20} \cdot 3^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(1.02468\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{20} \cdot 3^{8} \cdot 7^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(24.75197287\)
\(L(\frac12)\) \(\approx\) \(24.75197287\)
\(L(1)\) 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$C_1$ \( ( 1 - T )^{4} \)
3 \( 1 \)
7$C_2^2$ \( 1 + T^{4} \)
good5$C_4\times C_2$ \( 1 + T^{8} \)
11$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
13$C_4\times C_2$ \( 1 + T^{8} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
19$C_4\times C_2$ \( 1 + T^{8} \)
23$C_2^2$ \( ( 1 + T^{4} )^{2} \)
29$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
31$C_2$ \( ( 1 + T^{2} )^{4} \)
37$C_1$$\times$$C_2^2$ \( ( 1 + T )^{4}( 1 + T^{4} ) \)
41$C_2^2$ \( ( 1 + T^{4} )^{2} \)
43$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
47$C_2$ \( ( 1 + T^{2} )^{4} \)
53$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
59$C_4\times C_2$ \( 1 + T^{8} \)
61$C_4\times C_2$ \( 1 + T^{8} \)
67$C_1$$\times$$C_2^2$ \( ( 1 + T )^{4}( 1 + T^{4} ) \)
71$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \)
73$C_2^2$ \( ( 1 + T^{4} )^{2} \)
79$C_2^2$ \( ( 1 + T^{4} )^{2} \)
83$C_4\times C_2$ \( 1 + T^{8} \)
89$C_2^2$ \( ( 1 + T^{4} )^{2} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
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

−6.73286939350955501460688611893, −6.21090905980475640858097136447, −6.09717798310453723080261266767, −6.07447399905347671047239071367, −5.81845693527872974969857032615, −5.45805981228109190225702640315, −5.39104287709016801832442784285, −5.29542797413871424001298801107, −5.03613554830650975941025802821, −4.58067041896896796114031429709, −4.57702720483492046667821306302, −4.50485158055025781113533528055, −4.23431157981915165963925060638, −3.77034675330082694232536658859, −3.62238035867631709600739697164, −3.57928445123212443906186708855, −3.42473850030887990049724625216, −2.77389108253624107216387536119, −2.75556111537211227883954330145, −2.73522286090890548931560528823, −2.48715432468361199316269978137, −1.68764139334497581913424624736, −1.66567673276993566850042753003, −1.56054297437269078064248583873, −1.34491021891440770966109071727, 1.34491021891440770966109071727, 1.56054297437269078064248583873, 1.66567673276993566850042753003, 1.68764139334497581913424624736, 2.48715432468361199316269978137, 2.73522286090890548931560528823, 2.75556111537211227883954330145, 2.77389108253624107216387536119, 3.42473850030887990049724625216, 3.57928445123212443906186708855, 3.62238035867631709600739697164, 3.77034675330082694232536658859, 4.23431157981915165963925060638, 4.50485158055025781113533528055, 4.57702720483492046667821306302, 4.58067041896896796114031429709, 5.03613554830650975941025802821, 5.29542797413871424001298801107, 5.39104287709016801832442784285, 5.45805981228109190225702640315, 5.81845693527872974969857032615, 6.07447399905347671047239071367, 6.09717798310453723080261266767, 6.21090905980475640858097136447, 6.73286939350955501460688611893

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