Properties

Label 8-1440e4-1.1-c0e4-0-1
Degree $8$
Conductor $4.300\times 10^{12}$
Sign $1$
Analytic cond. $0.266734$
Root an. cond. $0.847734$
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  + 4·7-s + 8·49-s + 4·73-s − 4·97-s − 4·103-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 + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯
L(s)  = 1  + 4·7-s + 8·49-s + 4·73-s − 4·97-s − 4·103-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 + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{20} \cdot 3^{8} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(0.266734\)
Root analytic conductor: \(0.847734\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1440} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{20} \cdot 3^{8} \cdot 5^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.063683439\)
\(L(\frac12)\) \(\approx\) \(2.063683439\)
\(L(1)\) 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 \( 1 \)
5$C_2^2$ \( 1 + T^{4} \)
good7$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \)
11$C_2^2$ \( ( 1 + T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + T^{4} )^{2} \)
19$C_2$ \( ( 1 + T^{2} )^{4} \)
23$C_2^2$ \( ( 1 + T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + T^{4} )^{2} \)
31$C_2$ \( ( 1 + T^{2} )^{4} \)
37$C_2^2$ \( ( 1 + T^{4} )^{2} \)
41$C_2$ \( ( 1 + T^{2} )^{4} \)
43$C_2^2$ \( ( 1 + T^{4} )^{2} \)
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_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
67$C_2^2$ \( ( 1 + T^{4} )^{2} \)
71$C_2$ \( ( 1 + T^{2} )^{4} \)
73$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
83$C_2^2$ \( ( 1 + T^{4} )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
97$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \)
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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

−7.23877932339360326864512391072, −6.62240641491018134764554524249, −6.57920763497318230169131748881, −6.42423265312709727222128965060, −6.31141017574478206698481934846, −5.72530047700319781805796430884, −5.49478325187509058579704908580, −5.36110192252952478220764486783, −5.28060923241097287067246004484, −5.03784283417454863002743097183, −4.82763904591644992128280019524, −4.68335828631591790826189762781, −4.42053086846019867613949769744, −4.05074312259481213836843855944, −3.89937790894559906117786684628, −3.69127556985221512978170843970, −3.61721760789822778265946992051, −2.69607859618904268413154831238, −2.65384282431495531795497896803, −2.57850643967214927653842104658, −2.18961229420917321246268900238, −1.67726755341189007465008378223, −1.51646690258066704947517234067, −1.36293816973987177981554515946, −1.01192412048972559087686914864, 1.01192412048972559087686914864, 1.36293816973987177981554515946, 1.51646690258066704947517234067, 1.67726755341189007465008378223, 2.18961229420917321246268900238, 2.57850643967214927653842104658, 2.65384282431495531795497896803, 2.69607859618904268413154831238, 3.61721760789822778265946992051, 3.69127556985221512978170843970, 3.89937790894559906117786684628, 4.05074312259481213836843855944, 4.42053086846019867613949769744, 4.68335828631591790826189762781, 4.82763904591644992128280019524, 5.03784283417454863002743097183, 5.28060923241097287067246004484, 5.36110192252952478220764486783, 5.49478325187509058579704908580, 5.72530047700319781805796430884, 6.31141017574478206698481934846, 6.42423265312709727222128965060, 6.57920763497318230169131748881, 6.62240641491018134764554524249, 7.23877932339360326864512391072

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