Properties

Label 8-192e4-1.1-c3e4-0-3
Degree $8$
Conductor $1358954496$
Sign $1$
Analytic cond. $16469.0$
Root an. cond. $3.36576$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 18·9-s + 216·17-s + 476·25-s − 1.17e3·41-s − 196·49-s − 4.02e3·73-s + 243·81-s − 5.92e3·89-s + 7.28e3·97-s − 1.56e3·113-s + 716·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 3.88e3·153-s + 157-s + 163-s + 167-s + 5.33e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 2/3·9-s + 3.08·17-s + 3.80·25-s − 4.47·41-s − 4/7·49-s − 6.45·73-s + 1/3·81-s − 7.06·89-s + 7.62·97-s − 1.29·113-s + 0.537·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s − 2.05·153-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 2.42·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(16469.0\)
Root analytic conductor: \(3.36576\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 3^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(1.712290147\)
\(L(\frac12)\) \(\approx\) \(1.712290147\)
\(L(\frac{5}{2})\) 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$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
good5$C_2^2$ \( ( 1 - 238 T^{2} + p^{6} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 + 2 p^{2} T^{2} + p^{6} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 358 T^{2} + p^{6} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 2666 T^{2} + p^{6} T^{4} )^{2} \)
17$C_2$ \( ( 1 - 54 T + p^{3} T^{2} )^{4} \)
19$C_2^2$ \( ( 1 - 13702 T^{2} + p^{6} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 5666 T^{2} + p^{6} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 22270 T^{2} + p^{6} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 56114 T^{2} + p^{6} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 + 4726 T^{2} + p^{6} T^{4} )^{2} \)
41$C_2$ \( ( 1 + 294 T + p^{3} T^{2} )^{4} \)
43$C_2^2$ \( ( 1 - 123670 T^{2} + p^{6} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 48146 T^{2} + p^{6} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + 256946 T^{2} + p^{6} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 347254 T^{2} + p^{6} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 445850 T^{2} + p^{6} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 207142 T^{2} + p^{6} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 715774 T^{2} + p^{6} T^{4} )^{2} \)
73$C_2$ \( ( 1 + 1006 T + p^{3} T^{2} )^{4} \)
79$C_2^2$ \( ( 1 - 811150 T^{2} + p^{6} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 625174 T^{2} + p^{6} T^{4} )^{2} \)
89$C_2$ \( ( 1 + 1482 T + p^{3} T^{2} )^{4} \)
97$C_2$ \( ( 1 - 1822 T + p^{3} T^{2} )^{4} \)
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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

−8.842873712942778089841656892782, −8.447021118912845150489565926778, −8.297502204857221619160732317865, −7.79097429715000858511811352523, −7.74016892317729523793644520842, −7.16634662060996807894710402171, −7.04212313399442265602401706554, −6.98598970870768570764569196367, −6.36508021614066942391817636056, −6.31825646262571470144875020913, −5.62342742433537457305644356755, −5.57711194019405731154985331277, −5.45267556693726555884314250413, −4.86743375806326187067162169522, −4.66810372483814567510359672167, −4.55380105749821572389403599080, −3.75582166561291065245240375078, −3.35834373625043182989910255615, −3.20261935189415211060644030745, −2.94003332498992884927593853240, −2.69348918937598456249345726662, −1.73050350193356663336008338961, −1.24878618868391497694858730949, −1.23184531978919610774676607727, −0.27466880455220668469305860575, 0.27466880455220668469305860575, 1.23184531978919610774676607727, 1.24878618868391497694858730949, 1.73050350193356663336008338961, 2.69348918937598456249345726662, 2.94003332498992884927593853240, 3.20261935189415211060644030745, 3.35834373625043182989910255615, 3.75582166561291065245240375078, 4.55380105749821572389403599080, 4.66810372483814567510359672167, 4.86743375806326187067162169522, 5.45267556693726555884314250413, 5.57711194019405731154985331277, 5.62342742433537457305644356755, 6.31825646262571470144875020913, 6.36508021614066942391817636056, 6.98598970870768570764569196367, 7.04212313399442265602401706554, 7.16634662060996807894710402171, 7.74016892317729523793644520842, 7.79097429715000858511811352523, 8.297502204857221619160732317865, 8.447021118912845150489565926778, 8.842873712942778089841656892782

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