Properties

Label 8-3744e4-1.1-c0e4-0-5
Degree $8$
Conductor $1.965\times 10^{14}$
Sign $1$
Analytic cond. $12.1891$
Root an. cond. $1.36693$
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  + 4·13-s − 4·25-s − 4·49-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-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 + 239-s + 241-s + ⋯
L(s)  = 1  + 4·13-s − 4·25-s − 4·49-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-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 + 239-s + 241-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{20} \cdot 3^{8} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(12.1891\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{20} \cdot 3^{8} \cdot 13^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.711175208\)
\(L(\frac12)\) \(\approx\) \(1.711175208\)
\(L(1)\) 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 \( 1 \)
13$C_1$ \( ( 1 - T )^{4} \)
good5$C_2$ \( ( 1 + T^{2} )^{4} \)
7$C_2$ \( ( 1 + T^{2} )^{4} \)
11$C_2^2$ \( ( 1 + T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + T^{4} )^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
29$C_2^2$ \( ( 1 + T^{4} )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
41$C_2$ \( ( 1 + T^{2} )^{4} \)
43$C_2$ \( ( 1 + T^{2} )^{4} \)
47$C_2^2$ \( ( 1 + T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + T^{4} )^{2} \)
61$C_2$ \( ( 1 + T^{2} )^{4} \)
67$C_2$ \( ( 1 + T^{2} )^{4} \)
71$C_2^2$ \( ( 1 + T^{4} )^{2} \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
79$C_2$ \( ( 1 + T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + T^{4} )^{2} \)
89$C_2$ \( ( 1 + T^{2} )^{4} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
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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

−6.33248571801848505585409516999, −5.89186004332812918733570909831, −5.88497423548763766168385900960, −5.73318104521047876448059632311, −5.46669440232052679027970366996, −5.29308371915132651551587751796, −5.21063137483275099168203687998, −4.72516415845926324803587653670, −4.44749028957424214909209794350, −4.25836106705058024385163749480, −4.16999996962144255646405830890, −4.12527497364545517259097100481, −3.67461667684720974056528045528, −3.43442285769496290198143521626, −3.37128124558202806589129757407, −3.26218948561992834014558808141, −3.12747041926156170629589083624, −2.49891886504529678394497968118, −2.40016820941047323624451859989, −1.79049215104578097119088190204, −1.76662863656922396801212754650, −1.71997506190771126393735826332, −1.40337545614928949919051619408, −0.963887853513604646285638321325, −0.49703993500675149397681882914, 0.49703993500675149397681882914, 0.963887853513604646285638321325, 1.40337545614928949919051619408, 1.71997506190771126393735826332, 1.76662863656922396801212754650, 1.79049215104578097119088190204, 2.40016820941047323624451859989, 2.49891886504529678394497968118, 3.12747041926156170629589083624, 3.26218948561992834014558808141, 3.37128124558202806589129757407, 3.43442285769496290198143521626, 3.67461667684720974056528045528, 4.12527497364545517259097100481, 4.16999996962144255646405830890, 4.25836106705058024385163749480, 4.44749028957424214909209794350, 4.72516415845926324803587653670, 5.21063137483275099168203687998, 5.29308371915132651551587751796, 5.46669440232052679027970366996, 5.73318104521047876448059632311, 5.88497423548763766168385900960, 5.89186004332812918733570909831, 6.33248571801848505585409516999

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