Properties

Label 8-24e8-1.1-c3e4-0-9
Degree $8$
Conductor $110075314176$
Sign $1$
Analytic cond. $1.33399\times 10^{6}$
Root an. cond. $5.82967$
Motivic weight $3$
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  − 216·17-s + 476·25-s + 1.17e3·41-s − 196·49-s − 4.02e3·73-s + 5.92e3·89-s + 7.28e3·97-s + 1.56e3·113-s + 716·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5.33e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  − 3.08·17-s + 3.80·25-s + 4.47·41-s − 4/7·49-s − 6.45·73-s + 7.06·89-s + 7.62·97-s + 1.29·113-s + 0.537·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 2.42·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(1.33399\times 10^{6}\)
Root analytic conductor: \(5.82967\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 3^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(6.849160590\)
\(L(\frac12)\) \(\approx\) \(6.849160590\)
\(L(\frac{5}{2})\) 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$C_2^2$ \( ( 1 - 238 T^{2} + p^{6} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 + 2 p^{2} T^{2} + p^{6} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 358 T^{2} + p^{6} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 2666 T^{2} + p^{6} T^{4} )^{2} \)
17$C_2$ \( ( 1 + 54 T + p^{3} T^{2} )^{4} \)
19$C_2^2$ \( ( 1 - 13702 T^{2} + p^{6} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 5666 T^{2} + p^{6} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 22270 T^{2} + p^{6} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 56114 T^{2} + p^{6} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 + 4726 T^{2} + p^{6} T^{4} )^{2} \)
41$C_2$ \( ( 1 - 294 T + p^{3} T^{2} )^{4} \)
43$C_2^2$ \( ( 1 - 123670 T^{2} + p^{6} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 48146 T^{2} + p^{6} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + 256946 T^{2} + p^{6} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 347254 T^{2} + p^{6} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 445850 T^{2} + p^{6} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 207142 T^{2} + p^{6} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 715774 T^{2} + p^{6} T^{4} )^{2} \)
73$C_2$ \( ( 1 + 1006 T + p^{3} T^{2} )^{4} \)
79$C_2^2$ \( ( 1 - 811150 T^{2} + p^{6} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 625174 T^{2} + p^{6} T^{4} )^{2} \)
89$C_2$ \( ( 1 - 1482 T + p^{3} T^{2} )^{4} \)
97$C_2$ \( ( 1 - 1822 T + p^{3} T^{2} )^{4} \)
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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

−7.31945038217256939267313705900, −7.07041787902609996027711067623, −6.87561346093139081682795401885, −6.53193620338873006370821076968, −6.27631782757155007668327494359, −6.11132725978063392415795217119, −6.09766040698839433911227097284, −5.64635638138415611073676200182, −5.38231592395590540146583323258, −4.68495035856480633254807283262, −4.68168202979267780797657257751, −4.66864762178419994200437151574, −4.49930217344175819805746519605, −4.29023293072434679065508450379, −3.47413123486996255573483745036, −3.44629429571768824482172349234, −3.24653178779226834919270950067, −2.64614978469785533272485355759, −2.44693025787220538262170362356, −2.38083665624231994010067269951, −1.82816391527957270050066060978, −1.55632438754891291247340403404, −0.76867519395063296878116060024, −0.67351929185989974637924907125, −0.51763904594110609012799638514, 0.51763904594110609012799638514, 0.67351929185989974637924907125, 0.76867519395063296878116060024, 1.55632438754891291247340403404, 1.82816391527957270050066060978, 2.38083665624231994010067269951, 2.44693025787220538262170362356, 2.64614978469785533272485355759, 3.24653178779226834919270950067, 3.44629429571768824482172349234, 3.47413123486996255573483745036, 4.29023293072434679065508450379, 4.49930217344175819805746519605, 4.66864762178419994200437151574, 4.68168202979267780797657257751, 4.68495035856480633254807283262, 5.38231592395590540146583323258, 5.64635638138415611073676200182, 6.09766040698839433911227097284, 6.11132725978063392415795217119, 6.27631782757155007668327494359, 6.53193620338873006370821076968, 6.87561346093139081682795401885, 7.07041787902609996027711067623, 7.31945038217256939267313705900

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