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Properties

Label 8-2040e4-1.1-c0e4-0-1
Degree $8$
Conductor $1.732\times 10^{13}$
Sign $1$
Analytic cond. $1.07435$
Root an. cond. $1.00900$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s + 3·16-s − 4·23-s + 4·31-s − 8·47-s − 4·64-s − 4·79-s − 81-s + 8·92-s + 4·113-s − 8·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 16·188-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  − 2·4-s + 3·16-s − 4·23-s + 4·31-s − 8·47-s − 4·64-s − 4·79-s − 81-s + 8·92-s + 4·113-s − 8·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 16·188-s + 191-s + 193-s + 197-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 17^{4}\)
Sign: $1$
Analytic conductor: \(1.07435\)
Root analytic conductor: \(1.00900\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 17^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1646432143\)
\(L(\frac12)\) \(\approx\) \(0.1646432143\)
\(L(1)\) 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$C_2$ \( ( 1 + T^{2} )^{2} \)
3$C_2^2$ \( 1 + T^{4} \)
5$C_2^2$ \( 1 + T^{4} \)
17$C_2$ \( ( 1 + T^{2} )^{2} \)
good7$C_2^2$ \( ( 1 + T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + T^{4} )^{2} \)
19$C_2$ \( ( 1 + T^{2} )^{4} \)
23$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \)
29$C_2^2$ \( ( 1 + T^{4} )^{2} \)
31$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \)
37$C_2^2$ \( ( 1 + T^{4} )^{2} \)
41$C_2^2$ \( ( 1 + T^{4} )^{2} \)
43$C_2^2$ \( ( 1 + T^{4} )^{2} \)
47$C_1$ \( ( 1 + T )^{8} \)
53$C_2$ \( ( 1 + T^{2} )^{4} \)
59$C_2^2$ \( ( 1 + T^{4} )^{2} \)
61$C_2^2$ \( ( 1 + T^{4} )^{2} \)
67$C_2^2$ \( ( 1 + T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + T^{4} )^{2} \)
73$C_2^2$ \( ( 1 + T^{4} )^{2} \)
79$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \)
83$C_2$ \( ( 1 + T^{2} )^{4} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
97$C_2^2$ \( ( 1 + T^{4} )^{2} \)
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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

−6.57935498276410435728835911223, −6.48648700368981223430835552726, −6.34412689990504437756198876556, −6.04156209771904457113716678693, −5.70674229558274230796762933026, −5.68699881717952081235481234994, −5.60875504943087291659606758862, −5.04410494092845490740709029680, −4.85172087290494664931167109406, −4.70604454540389896992295299911, −4.52672796658609031094945337967, −4.51092979352477532666482874894, −4.20218366760448872586251997121, −3.89521570406813686883207922575, −3.77514310554604001995538009842, −3.29924914049728981776873363337, −3.17741951227808935812566039568, −2.98446240989403607976166261642, −2.92884014780591068026389085606, −2.20995232501758188853121378809, −1.87265361356554674802163745161, −1.85780675927043234313661992146, −1.24576847235319261336291968950, −1.18023444200388839132471701935, −0.22839067864915805882443768634, 0.22839067864915805882443768634, 1.18023444200388839132471701935, 1.24576847235319261336291968950, 1.85780675927043234313661992146, 1.87265361356554674802163745161, 2.20995232501758188853121378809, 2.92884014780591068026389085606, 2.98446240989403607976166261642, 3.17741951227808935812566039568, 3.29924914049728981776873363337, 3.77514310554604001995538009842, 3.89521570406813686883207922575, 4.20218366760448872586251997121, 4.51092979352477532666482874894, 4.52672796658609031094945337967, 4.70604454540389896992295299911, 4.85172087290494664931167109406, 5.04410494092845490740709029680, 5.60875504943087291659606758862, 5.68699881717952081235481234994, 5.70674229558274230796762933026, 6.04156209771904457113716678693, 6.34412689990504437756198876556, 6.48648700368981223430835552726, 6.57935498276410435728835911223

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