Properties

Label 8-2960e4-1.1-c0e4-0-4
Degree $8$
Conductor $7.677\times 10^{13}$
Sign $1$
Analytic cond. $4.76206$
Root an. cond. $1.21541$
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  + 4·5-s + 10·25-s − 4·37-s + 4·121-s + 20·125-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 − 16·185-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯
L(s)  = 1  + 4·5-s + 10·25-s − 4·37-s + 4·121-s + 20·125-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 − 16·185-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 5^{4} \cdot 37^{4}\)
Sign: $1$
Analytic conductor: \(4.76206\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 5^{4} \cdot 37^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.423029684\)
\(L(\frac12)\) \(\approx\) \(4.423029684\)
\(L(1)\) 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 \( 1 \)
5$C_1$ \( ( 1 - T )^{4} \)
37$C_1$ \( ( 1 + T )^{4} \)
good3$C_4\times C_2$ \( 1 + T^{8} \)
7$C_4\times C_2$ \( 1 + T^{8} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
13$C_2^2$ \( ( 1 + T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + T^{4} )^{2} \)
19$C_4\times C_2$ \( 1 + T^{8} \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
31$C_4\times C_2$ \( 1 + T^{8} \)
41$C_2$ \( ( 1 + T^{2} )^{4} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
47$C_4\times C_2$ \( 1 + T^{8} \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
59$C_4\times C_2$ \( 1 + T^{8} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
67$C_4\times C_2$ \( 1 + T^{8} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
79$C_4\times C_2$ \( 1 + T^{8} \)
83$C_4\times C_2$ \( 1 + T^{8} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
97$C_2$ \( ( 1 + T^{2} )^{4} \)
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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

−6.53522734140423872577700925148, −6.01075568525397572502047720062, −5.92320746130541755723837714914, −5.83720077064645593394904591322, −5.62123904542393599393017546864, −5.44245233462647310503391473304, −5.15977706990346324129702330577, −5.06849577797753591003416359592, −4.90784432372750134413916796277, −4.86240851631867223404589228158, −4.26190538940091320547791134437, −4.22178822479862124011881888858, −4.01274592519818154614592136095, −3.38901458300727354364851175705, −3.22874265315570078960190221946, −3.20614281579893070705733529036, −3.00919050702118034218511358285, −2.51772256646198091816166950782, −2.45337403319714641053160837473, −1.95927319928669233105477970803, −1.92791148320480093421018438942, −1.73994129204124647986390066100, −1.63055450467082425521258220346, −1.02540076986606431174084601730, −0.874563945516450198223985091432, 0.874563945516450198223985091432, 1.02540076986606431174084601730, 1.63055450467082425521258220346, 1.73994129204124647986390066100, 1.92791148320480093421018438942, 1.95927319928669233105477970803, 2.45337403319714641053160837473, 2.51772256646198091816166950782, 3.00919050702118034218511358285, 3.20614281579893070705733529036, 3.22874265315570078960190221946, 3.38901458300727354364851175705, 4.01274592519818154614592136095, 4.22178822479862124011881888858, 4.26190538940091320547791134437, 4.86240851631867223404589228158, 4.90784432372750134413916796277, 5.06849577797753591003416359592, 5.15977706990346324129702330577, 5.44245233462647310503391473304, 5.62123904542393599393017546864, 5.83720077064645593394904591322, 5.92320746130541755723837714914, 6.01075568525397572502047720062, 6.53522734140423872577700925148

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