Properties

Label 8-1536e4-1.1-c0e4-0-1
Degree $8$
Conductor $5.566\times 10^{12}$
Sign $1$
Analytic cond. $0.345297$
Root an. cond. $0.875536$
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  − 2·9-s + 3·81-s − 8·97-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-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 + ⋯
L(s)  = 1  − 2·9-s + 3·81-s − 8·97-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{36} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(0.345297\)
Root analytic conductor: \(0.875536\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1536} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{36} \cdot 3^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6172872165\)
\(L(\frac12)\) \(\approx\) \(0.6172872165\)
\(L(1)\) 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$C_2$ \( ( 1 + T^{2} )^{2} \)
good5$C_2^2$ \( ( 1 + T^{4} )^{2} \)
7$C_2^2$ \( ( 1 + T^{4} )^{2} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
13$C_2$ \( ( 1 + T^{2} )^{4} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
19$C_2$ \( ( 1 + T^{2} )^{4} \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
29$C_2^2$ \( ( 1 + T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + T^{4} )^{2} \)
37$C_2$ \( ( 1 + T^{2} )^{4} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
43$C_2$ \( ( 1 + T^{2} )^{4} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
53$C_2^2$ \( ( 1 + T^{4} )^{2} \)
59$C_2$ \( ( 1 + T^{2} )^{4} \)
61$C_2$ \( ( 1 + T^{2} )^{4} \)
67$C_2$ \( ( 1 + T^{2} )^{4} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
73$C_2$ \( ( 1 + T^{2} )^{4} \)
79$C_2^2$ \( ( 1 + T^{4} )^{2} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
97$C_1$ \( ( 1 + T )^{8} \)
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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

−6.79086485975441440948786605513, −6.73824302184038454308079115223, −6.70469351939790803109367351248, −6.06721125677170315531463711802, −6.02447825685202347727637113729, −6.01164636642245522598555766092, −5.68953661969341963097242831986, −5.32786036996171564093830852157, −5.30535955489511374760814749431, −5.15416656696291014008391082304, −4.76938348563243513634370399072, −4.52003089030410023500199749084, −4.19659847462567164862127246443, −4.14019518564257500800259096714, −3.77209691869147099429817644126, −3.38582658671152759664466590645, −3.33399801338009607358460732162, −3.05760073405233980040513067642, −2.59566197107718367984931429705, −2.48685273683211512943679477018, −2.47918794121677231774986272014, −1.83323515450409766242867810768, −1.50900546726229805037942865789, −1.21128091372142984625068142395, −0.47723753458551672793786135729, 0.47723753458551672793786135729, 1.21128091372142984625068142395, 1.50900546726229805037942865789, 1.83323515450409766242867810768, 2.47918794121677231774986272014, 2.48685273683211512943679477018, 2.59566197107718367984931429705, 3.05760073405233980040513067642, 3.33399801338009607358460732162, 3.38582658671152759664466590645, 3.77209691869147099429817644126, 4.14019518564257500800259096714, 4.19659847462567164862127246443, 4.52003089030410023500199749084, 4.76938348563243513634370399072, 5.15416656696291014008391082304, 5.30535955489511374760814749431, 5.32786036996171564093830852157, 5.68953661969341963097242831986, 6.01164636642245522598555766092, 6.02447825685202347727637113729, 6.06721125677170315531463711802, 6.70469351939790803109367351248, 6.73824302184038454308079115223, 6.79086485975441440948786605513

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