Properties

Label 8-1536e4-1.1-c1e4-0-17
Degree $8$
Conductor $5.566\times 10^{12}$
Sign $1$
Analytic cond. $22629.4$
Root an. cond. $3.50214$
Motivic weight $1$
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  + 4·3-s + 6·9-s − 8·25-s − 4·27-s − 24·59-s + 32·73-s − 32·75-s − 37·81-s + 32·83-s − 24·97-s + 8·107-s − 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-s + 173-s − 96·177-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  + 2.30·3-s + 2·9-s − 8/5·25-s − 0.769·27-s − 3.12·59-s + 3.74·73-s − 3.69·75-s − 4.11·81-s + 3.51·83-s − 2.43·97-s + 0.773·107-s − 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4/13·169-s + 0.0760·173-s − 7.21·177-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{36} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(22629.4\)
Root analytic conductor: \(3.50214\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{36} \cdot 3^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(6.570969056\)
\(L(\frac12)\) \(\approx\) \(6.570969056\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
good5$C_2^2$ \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{4} \)
13$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{4} \)
29$C_2^2$ \( ( 1 - 44 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 48 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 - p T^{2} )^{4} \)
47$C_2^2$ \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 92 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
61$C_2^2$ \( ( 1 + 94 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 126 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 - 8 T + p T^{2} )^{4} \)
79$C_2^2$ \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2$ \( ( 1 - 8 T + p T^{2} )^{4} \)
89$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
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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

−6.73600682325751524153207315011, −6.51027211795705530359336841459, −6.40482307731168435037109958820, −6.17949911073715924508370727998, −5.81674348829347888986346877873, −5.78872670000539461921160668681, −5.30094741434198405656400884057, −5.21626962257089635237849400573, −4.94800836426980709928146557675, −4.86745999378051313050770293990, −4.29041125003802323180891628170, −4.08113599331139298438808498078, −3.95045015685627879453431620080, −3.76302733385025352899642647357, −3.65624692964634223975133314735, −3.14881131853906515460315136230, −2.92662363416923049994239158982, −2.84717396316866177956793406235, −2.67355403940169630331984932990, −2.19238968509742337452242520391, −1.82405520652112964728782513811, −1.71484068278769476674720908970, −1.70091171195328145363765915766, −0.74338130101622027691977053287, −0.44054685506982448859455459342, 0.44054685506982448859455459342, 0.74338130101622027691977053287, 1.70091171195328145363765915766, 1.71484068278769476674720908970, 1.82405520652112964728782513811, 2.19238968509742337452242520391, 2.67355403940169630331984932990, 2.84717396316866177956793406235, 2.92662363416923049994239158982, 3.14881131853906515460315136230, 3.65624692964634223975133314735, 3.76302733385025352899642647357, 3.95045015685627879453431620080, 4.08113599331139298438808498078, 4.29041125003802323180891628170, 4.86745999378051313050770293990, 4.94800836426980709928146557675, 5.21626962257089635237849400573, 5.30094741434198405656400884057, 5.78872670000539461921160668681, 5.81674348829347888986346877873, 6.17949911073715924508370727998, 6.40482307731168435037109958820, 6.51027211795705530359336841459, 6.73600682325751524153207315011

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