Properties

Label 8-12e12-1.1-c0e4-0-2
Degree $8$
Conductor $8.916\times 10^{12}$
Sign $1$
Analytic cond. $0.553099$
Root an. cond. $0.928646$
Motivic weight $0$
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  − 2·25-s + 2·49-s + 4·73-s − 4·97-s + 2·121-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 + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯
L(s)  = 1  − 2·25-s + 2·49-s + 4·73-s − 4·97-s + 2·121-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 + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 3^{12}\)
Sign: $1$
Analytic conductor: \(0.553099\)
Root analytic conductor: \(0.928646\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1728} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 3^{12} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.037810246\)
\(L(\frac12)\) \(\approx\) \(1.037810246\)
\(L(1)\) 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 \( 1 \)
3 \( 1 \)
good5$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
7$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
17$C_2$ \( ( 1 + T^{2} )^{4} \)
19$C_2$ \( ( 1 + T^{2} )^{4} \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
31$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
41$C_2$ \( ( 1 + T^{2} )^{4} \)
43$C_2$ \( ( 1 + T^{2} )^{4} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
53$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
59$C_2$ \( ( 1 + T^{2} )^{4} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
67$C_2$ \( ( 1 + T^{2} )^{4} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
73$C_2$ \( ( 1 - T + T^{2} )^{4} \)
79$C_2$ \( ( 1 + T^{2} )^{4} \)
83$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
89$C_2$ \( ( 1 + T^{2} )^{4} \)
97$C_2$ \( ( 1 + T + T^{2} )^{4} \)
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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

−6.96554693003475030939259925085, −6.69667009869626762415287951612, −6.56384369398420986721041059111, −6.01058269903611076093421220034, −5.85764038646417197966560838426, −5.85216650966682629769221314303, −5.71458768147761223305377234574, −5.36461389378710630803584534790, −5.10648996493238634169990687584, −5.01468287539011956956548146856, −4.46875602335939669220137224221, −4.46256558369872648350477891297, −4.35252477490520141037581438516, −3.73471513026515533627593127315, −3.72500822319344091911680340217, −3.69020245367746803452221434075, −3.34216952906879499504036469782, −2.87473423681427240939004859718, −2.65024315508821018248780196458, −2.38766893729123080263566463660, −2.15091601406651104804112614109, −1.90124744269204405741049955552, −1.48913112638116237446730664632, −1.12668990585797744293638940924, −0.60370191646727068006050159353, 0.60370191646727068006050159353, 1.12668990585797744293638940924, 1.48913112638116237446730664632, 1.90124744269204405741049955552, 2.15091601406651104804112614109, 2.38766893729123080263566463660, 2.65024315508821018248780196458, 2.87473423681427240939004859718, 3.34216952906879499504036469782, 3.69020245367746803452221434075, 3.72500822319344091911680340217, 3.73471513026515533627593127315, 4.35252477490520141037581438516, 4.46256558369872648350477891297, 4.46875602335939669220137224221, 5.01468287539011956956548146856, 5.10648996493238634169990687584, 5.36461389378710630803584534790, 5.71458768147761223305377234574, 5.85216650966682629769221314303, 5.85764038646417197966560838426, 6.01058269903611076093421220034, 6.56384369398420986721041059111, 6.69667009869626762415287951612, 6.96554693003475030939259925085

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