Properties

Label 8-96e8-1.1-c1e4-0-12
Degree $8$
Conductor $7.214\times 10^{15}$
Sign $1$
Analytic cond. $2.93277\times 10^{7}$
Root an. cond. $8.57846$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 24·19-s + 4·25-s − 24·43-s − 24·49-s + 32·67-s − 48·73-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 48·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯
L(s)  = 1  − 5.50·19-s + 4/5·25-s − 3.65·43-s − 3.42·49-s + 3.90·67-s − 5.61·73-s + 4/11·121-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 − 3.69·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{40} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(2.93277\times 10^{7}\)
Root analytic conductor: \(8.57846\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{40} \cdot 3^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 + 12 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + 24 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
23$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 60 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 24 T^{2} + p^{2} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
47$C_2^2$ \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + 94 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 + 72 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{4} \)
71$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 + 12 T + p T^{2} )^{4} \)
79$C_2^2$ \( ( 1 - 84 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + 82 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2$ \( ( 1 + p T^{2} )^{4} \)
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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

−6.02489970593201189551814219166, −5.54245343299070255497200109443, −5.30373878340142683612726101177, −5.11233659923648313312959291066, −5.00091914844886294912042776864, −4.90317799983325326071561292539, −4.61979646511995695550295598476, −4.52928333662340207674555349196, −4.43921837081396125265569621880, −4.02603117839198979773649161646, −3.84127785528522437124290407757, −3.80165549710596219619860832081, −3.79270785389549050535519405814, −3.34391112251201480845146288156, −3.03333064259171931554732589402, −2.87895194945816899619714765606, −2.85105861132253061164080104005, −2.51279105978183932680662705842, −2.10851374822822283464534646770, −2.06487761096570520188358102416, −1.99872516189706137765918691194, −1.60174049026904257438251011498, −1.52752812373692569625809622502, −1.14769649384116414293320463501, −0.900013525849858742360923892661, 0, 0, 0, 0, 0.900013525849858742360923892661, 1.14769649384116414293320463501, 1.52752812373692569625809622502, 1.60174049026904257438251011498, 1.99872516189706137765918691194, 2.06487761096570520188358102416, 2.10851374822822283464534646770, 2.51279105978183932680662705842, 2.85105861132253061164080104005, 2.87895194945816899619714765606, 3.03333064259171931554732589402, 3.34391112251201480845146288156, 3.79270785389549050535519405814, 3.80165549710596219619860832081, 3.84127785528522437124290407757, 4.02603117839198979773649161646, 4.43921837081396125265569621880, 4.52928333662340207674555349196, 4.61979646511995695550295598476, 4.90317799983325326071561292539, 5.00091914844886294912042776864, 5.11233659923648313312959291066, 5.30373878340142683612726101177, 5.54245343299070255497200109443, 6.02489970593201189551814219166

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