Properties

Label 8-1805e4-1.1-c0e4-0-1
Degree $8$
Conductor $1.061\times 10^{13}$
Sign $1$
Analytic cond. $0.658472$
Root an. cond. $0.949111$
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  + 2·5-s + 4·7-s + 16-s − 2·17-s + 2·23-s + 25-s + 8·35-s − 2·43-s − 2·47-s + 8·49-s − 2·73-s + 2·80-s + 81-s + 4·83-s − 4·85-s + 4·112-s + 4·115-s − 8·119-s − 4·121-s − 2·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  + 2·5-s + 4·7-s + 16-s − 2·17-s + 2·23-s + 25-s + 8·35-s − 2·43-s − 2·47-s + 8·49-s − 2·73-s + 2·80-s + 81-s + 4·83-s − 4·85-s + 4·112-s + 4·115-s − 8·119-s − 4·121-s − 2·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5^{4} \cdot 19^{8}\)
Sign: $1$
Analytic conductor: \(0.658472\)
Root analytic conductor: \(0.949111\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 5^{4} \cdot 19^{8} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.575505740\)
\(L(\frac12)\) \(\approx\) \(3.575505740\)
\(L(1)\) 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)$
bad5$C_2$ \( ( 1 - T + T^{2} )^{2} \)
19 \( 1 \)
good2$C_2^3$ \( 1 - T^{4} + T^{8} \)
3$C_2^3$ \( 1 - T^{4} + T^{8} \)
7$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \)
11$C_2$ \( ( 1 + T^{2} )^{4} \)
13$C_2^3$ \( 1 - T^{4} + T^{8} \)
17$C_2$$\times$$C_2^2$ \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
23$C_2$$\times$$C_2^2$ \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
29$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
31$C_2$ \( ( 1 + T^{2} )^{4} \)
37$C_2^2$ \( ( 1 + T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
43$C_2$$\times$$C_2^2$ \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
47$C_2$$\times$$C_2^2$ \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
53$C_2^3$ \( 1 - T^{4} + T^{8} \)
59$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
61$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
67$C_2^3$ \( 1 - T^{4} + T^{8} \)
71$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
73$C_2$$\times$$C_2^2$ \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
79$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
83$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \)
89$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
97$C_2^3$ \( 1 - T^{4} + T^{8} \)
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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

−6.96829873985163941699099799107, −6.39494738313924838814316052162, −6.35382912818920025163166578765, −6.15413903725315429259953379522, −5.95066718668048136819665674217, −5.73675234918728121096817945139, −5.39844900641203368948527819803, −5.13860173585919462425595207208, −5.12415546940069219469263240017, −4.83523607428323493651210226279, −4.72249620430373927388068272272, −4.61648928476780265969840442654, −4.58288894182124848905408330761, −3.84880483325986339020206104231, −3.80668435486843840396460578373, −3.40894672731197826436305030439, −3.30777871749459673314312383854, −2.69024534473089360619066240179, −2.41986216416759211326095713572, −2.32380069510543619947206085625, −1.95018773192803137022730556684, −1.90555224068989314331050168099, −1.36290705284242985829644560132, −1.28268344158568179228032519218, −1.23740987020273788573557037796, 1.23740987020273788573557037796, 1.28268344158568179228032519218, 1.36290705284242985829644560132, 1.90555224068989314331050168099, 1.95018773192803137022730556684, 2.32380069510543619947206085625, 2.41986216416759211326095713572, 2.69024534473089360619066240179, 3.30777871749459673314312383854, 3.40894672731197826436305030439, 3.80668435486843840396460578373, 3.84880483325986339020206104231, 4.58288894182124848905408330761, 4.61648928476780265969840442654, 4.72249620430373927388068272272, 4.83523607428323493651210226279, 5.12415546940069219469263240017, 5.13860173585919462425595207208, 5.39844900641203368948527819803, 5.73675234918728121096817945139, 5.95066718668048136819665674217, 6.15413903725315429259953379522, 6.35382912818920025163166578765, 6.39494738313924838814316052162, 6.96829873985163941699099799107

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