Properties

Label 8-63e8-1.1-c0e4-0-4
Degree $8$
Conductor $2.482\times 10^{14}$
Sign $1$
Analytic cond. $15.3940$
Root an. cond. $1.40740$
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  + 16-s + 4·25-s + 4·67-s − 4·79-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-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  + 16-s + 4·25-s + 4·67-s − 4·79-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-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

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{16} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(15.3940\)
Root analytic conductor: \(1.40740\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3969} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{16} \cdot 7^{8} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.190976027\)
\(L(\frac12)\) \(\approx\) \(2.190976027\)
\(L(1)\) 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)$
bad3 \( 1 \)
7 \( 1 \)
good2$C_2^3$ \( 1 - T^{4} + T^{8} \)
5$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
11$C_2^2$ \( ( 1 + T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
17$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
19$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + T^{4} )^{2} \)
29$C_2^3$ \( 1 - T^{4} + T^{8} \)
31$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
41$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
43$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
47$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
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$C_2$ \( ( 1 - T + T^{2} )^{4} \)
71$C_2^2$ \( ( 1 + T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
79$C_2$ \( ( 1 + T + T^{2} )^{4} \)
83$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
89$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
97$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
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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

−6.34968408213742728857899599366, −5.91509545421500459905379293666, −5.66186782385849580269429033617, −5.61943483591992481379901852248, −5.23056664176603041511809428049, −5.16374134524701658411467613424, −5.09630647132925392356481383977, −4.85539147311880201060511145956, −4.69188431773343411095073933560, −4.17903496516793863057858133539, −4.17745756437842316164910530509, −3.99701773176668348038927274079, −3.85379958417569450328375084324, −3.42373066292012790285283420771, −3.18974392243905934361556201746, −3.08351833145420381478160297391, −2.75948145275648521759044180893, −2.75181447751041419636730026925, −2.45641982340537286328424159620, −2.12283783571082670787491667006, −1.76577461545252241622351190108, −1.46795964380158488298119788714, −1.09195187999228471846735585429, −1.08454199789023276318843146659, −0.58779138915295839115903035492, 0.58779138915295839115903035492, 1.08454199789023276318843146659, 1.09195187999228471846735585429, 1.46795964380158488298119788714, 1.76577461545252241622351190108, 2.12283783571082670787491667006, 2.45641982340537286328424159620, 2.75181447751041419636730026925, 2.75948145275648521759044180893, 3.08351833145420381478160297391, 3.18974392243905934361556201746, 3.42373066292012790285283420771, 3.85379958417569450328375084324, 3.99701773176668348038927274079, 4.17745756437842316164910530509, 4.17903496516793863057858133539, 4.69188431773343411095073933560, 4.85539147311880201060511145956, 5.09630647132925392356481383977, 5.16374134524701658411467613424, 5.23056664176603041511809428049, 5.61943483591992481379901852248, 5.66186782385849580269429033617, 5.91509545421500459905379293666, 6.34968408213742728857899599366

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