Properties

Label 8-12e12-1.1-c0e4-0-1
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  − 4·25-s + 2·49-s − 4·73-s + 4·97-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-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  − 4·25-s + 2·49-s − 4·73-s + 4·97-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-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

\(L(\frac{1}{2})\) \(\approx\) \(0.7296794513\)
\(L(\frac12)\) \(\approx\) \(0.7296794513\)
\(L(1)\) 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$C_2$ \( ( 1 + T^{2} )^{4} \)
7$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
11$C_2$ \( ( 1 + T^{2} )^{4} \)
13$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
19$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
29$C_2$ \( ( 1 + T^{2} )^{4} \)
31$C_2$ \( ( 1 + T^{2} )^{4} \)
37$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
43$C_2$ \( ( 1 + T^{2} )^{4} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
53$C_2$ \( ( 1 + T^{2} )^{4} \)
59$C_2$ \( ( 1 + T^{2} )^{4} \)
61$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
67$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
73$C_2$ \( ( 1 + T + T^{2} )^{4} \)
79$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
83$C_2$ \( ( 1 + T^{2} )^{4} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
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

−6.99326595092749575480987268386, −6.45943846897056164066551919052, −6.36512830745111480021899452796, −6.10369930287206406779904534040, −6.00036035500522940460046212826, −5.90508806644719562779067785606, −5.51353532999741307869215834485, −5.45725681129531527842857408575, −5.26162929618539286342752890034, −4.69007782614118770680168609932, −4.65168210636875656123279438289, −4.59546122557385217629789820771, −4.08379331213660696936032823688, −3.87590463956139070015727376229, −3.80692219246757224978804995502, −3.63427805708805487319670145154, −3.25591163728466483437690393791, −2.99981303877586771403430792425, −2.48014192037986862179197496554, −2.38737495484767966059098893467, −2.31466449259314218333480453223, −1.72280138499637881989110314377, −1.53209581930402943391581363298, −1.27374928404865229311054650625, −0.46338051862625487952607940924, 0.46338051862625487952607940924, 1.27374928404865229311054650625, 1.53209581930402943391581363298, 1.72280138499637881989110314377, 2.31466449259314218333480453223, 2.38737495484767966059098893467, 2.48014192037986862179197496554, 2.99981303877586771403430792425, 3.25591163728466483437690393791, 3.63427805708805487319670145154, 3.80692219246757224978804995502, 3.87590463956139070015727376229, 4.08379331213660696936032823688, 4.59546122557385217629789820771, 4.65168210636875656123279438289, 4.69007782614118770680168609932, 5.26162929618539286342752890034, 5.45725681129531527842857408575, 5.51353532999741307869215834485, 5.90508806644719562779067785606, 6.00036035500522940460046212826, 6.10369930287206406779904534040, 6.36512830745111480021899452796, 6.45943846897056164066551919052, 6.99326595092749575480987268386

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