Properties

Label 8-2880e4-1.1-c1e4-0-11
Degree $8$
Conductor $6.880\times 10^{13}$
Sign $1$
Analytic cond. $279690.$
Root an. cond. $4.79550$
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  + 8·25-s − 28·49-s − 40·61-s + 80·109-s − 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 20·169-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  + 8/5·25-s − 4·49-s − 5.12·61-s + 7.66·109-s − 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.53·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 3^{8} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(279690.\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 3^{8} \cdot 5^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.819353097\)
\(L(\frac12)\) \(\approx\) \(1.819353097\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
good7$C_2$ \( ( 1 + p T^{2} )^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{4} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \)
17$C_2^2$ \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2$ \( ( 1 - p T^{2} )^{4} \)
23$C_2$ \( ( 1 - p T^{2} )^{4} \)
29$C_2^2$ \( ( 1 + 40 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 - p T^{2} )^{4} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( ( 1 - 80 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 + p T^{2} )^{4} \)
47$C_2$ \( ( 1 - p T^{2} )^{4} \)
53$C_2^2$ \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{4} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{4} \)
67$C_2$ \( ( 1 + p T^{2} )^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{4} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - p T^{2} )^{4} \)
83$C_2$ \( ( 1 - p T^{2} )^{4} \)
89$C_2^2$ \( ( 1 + 160 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2$ \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \)
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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

−6.13494086915020075409865432240, −6.06546148494935167464680147882, −5.97818947518039396037975198588, −5.72787517604859396369461257241, −5.31939642924819921024317444417, −5.03048874813687618606417851831, −4.90272712043525097576874807814, −4.83514446760996940854890576266, −4.62503756267678159935323704540, −4.46622408834133961545820480082, −4.30779398838333144128922440316, −3.78018416011000616958745541202, −3.57748127449401638057062749088, −3.44829073487706481654802646732, −3.21814698933365120573836652954, −3.08063301660503382020800422036, −2.73911539099751124753203876597, −2.60226033821790172229266870186, −2.12282744702941616672039386016, −2.04659483141336739451335690868, −1.42939919657847700598358526782, −1.37816120288386341083677580056, −1.34541006016027690662371859658, −0.55989771367511004188519884012, −0.25854935414654121881177736660, 0.25854935414654121881177736660, 0.55989771367511004188519884012, 1.34541006016027690662371859658, 1.37816120288386341083677580056, 1.42939919657847700598358526782, 2.04659483141336739451335690868, 2.12282744702941616672039386016, 2.60226033821790172229266870186, 2.73911539099751124753203876597, 3.08063301660503382020800422036, 3.21814698933365120573836652954, 3.44829073487706481654802646732, 3.57748127449401638057062749088, 3.78018416011000616958745541202, 4.30779398838333144128922440316, 4.46622408834133961545820480082, 4.62503756267678159935323704540, 4.83514446760996940854890576266, 4.90272712043525097576874807814, 5.03048874813687618606417851831, 5.31939642924819921024317444417, 5.72787517604859396369461257241, 5.97818947518039396037975198588, 6.06546148494935167464680147882, 6.13494086915020075409865432240

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