Properties

Label 8-12e12-1.1-c1e4-0-27
Degree $8$
Conductor $8.916\times 10^{12}$
Sign $1$
Analytic cond. $36247.9$
Root an. cond. $3.71458$
Motivic weight $1$
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  + 12·5-s − 12·13-s + 12·17-s + 74·25-s + 24·29-s + 6·41-s + 14·49-s − 24·61-s − 144·65-s − 28·73-s + 144·85-s + 24·89-s − 2·97-s + 12·101-s + 12·113-s − 13·121-s + 312·125-s + 127-s + 131-s + 137-s + 139-s + 288·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  + 5.36·5-s − 3.32·13-s + 2.91·17-s + 74/5·25-s + 4.45·29-s + 0.937·41-s + 2·49-s − 3.07·61-s − 17.8·65-s − 3.27·73-s + 15.6·85-s + 2.54·89-s − 0.203·97-s + 1.19·101-s + 1.12·113-s − 1.18·121-s + 27.9·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 23.9·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 3^{12}\)
Sign: $1$
Analytic conductor: \(36247.9\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1728} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 3^{12} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(19.09521166\)
\(L(\frac12)\) \(\approx\) \(19.09521166\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
good5$C_2^2$ \( ( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^3$ \( 1 + 13 T^{2} + 48 T^{4} + 13 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2^2$ \( ( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{4} \)
19$C_2^2$ \( ( 1 + 11 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^3$ \( 1 - 34 T^{2} + 627 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2^2$ \( ( 1 - 12 T + 77 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2^3$ \( 1 - 14 T^{2} - 765 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} \)
37$C_2^2$ \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \)
43$C_2^2$$\times$$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )( 1 + 83 T^{2} + p^{2} T^{4} ) \)
47$C_2^3$ \( 1 - 82 T^{2} + 4515 T^{4} - 82 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^2$ \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^3$ \( 1 + 37 T^{2} - 2112 T^{4} + 37 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2$ \( ( 1 - T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \)
67$C_2^2$$\times$$C_2^2$ \( ( 1 - 13 T^{2} + p^{2} T^{4} )( 1 + 122 T^{2} + p^{2} T^{4} ) \)
71$C_2^2$ \( ( 1 + 130 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 + 7 T + p T^{2} )^{4} \)
79$C_2^2$$\times$$C_2^2$ \( ( 1 + 11 T^{2} + p^{2} T^{4} )( 1 + 131 T^{2} + p^{2} T^{4} ) \)
83$C_2^3$ \( 1 + 22 T^{2} - 6405 T^{4} + 22 p^{2} T^{6} + p^{4} T^{8} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
97$C_2^2$ \( ( 1 + T - 96 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
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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

−6.65155783433132857816083728921, −6.27582942316648079635050548467, −6.08587138246801733960332167262, −6.02488369039259546153358990129, −5.73698433785611944257701998519, −5.65485000330765101271224939206, −5.32349377590537279337425916415, −5.28340774491140256173617070551, −5.15265450962864486281100532669, −4.74323412561662733424694718239, −4.54595742909872336223185203405, −4.44239531418290317411433848957, −4.37362792698862651615007050112, −3.36409807599021621800362521041, −3.33421790220116647836867597041, −2.95054744448761921632489044319, −2.81178918375364095660421719501, −2.53310844954150236041112997359, −2.52596997092205757437255558847, −2.09695836479419188043930573087, −2.00050092913382112877405957106, −1.56452015656206610589784415940, −1.30266658796319847891159881260, −1.03545967013686634439668251275, −0.68247852140964670359548011141, 0.68247852140964670359548011141, 1.03545967013686634439668251275, 1.30266658796319847891159881260, 1.56452015656206610589784415940, 2.00050092913382112877405957106, 2.09695836479419188043930573087, 2.52596997092205757437255558847, 2.53310844954150236041112997359, 2.81178918375364095660421719501, 2.95054744448761921632489044319, 3.33421790220116647836867597041, 3.36409807599021621800362521041, 4.37362792698862651615007050112, 4.44239531418290317411433848957, 4.54595742909872336223185203405, 4.74323412561662733424694718239, 5.15265450962864486281100532669, 5.28340774491140256173617070551, 5.32349377590537279337425916415, 5.65485000330765101271224939206, 5.73698433785611944257701998519, 6.02488369039259546153358990129, 6.08587138246801733960332167262, 6.27582942316648079635050548467, 6.65155783433132857816083728921

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