Properties

Degree $8$
Conductor $2.818\times 10^{13}$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·5-s + 12·13-s + 24·17-s + 32·25-s − 16·29-s + 12·37-s − 8·49-s − 16·53-s + 12·61-s − 96·65-s − 192·85-s − 16·97-s − 32·101-s − 28·109-s − 24·113-s − 104·125-s + 127-s + 131-s + 137-s + 139-s + 128·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 72·169-s + ⋯
L(s)  = 1  − 3.57·5-s + 3.32·13-s + 5.82·17-s + 32/5·25-s − 2.97·29-s + 1.97·37-s − 8/7·49-s − 2.19·53-s + 1.53·61-s − 11.9·65-s − 20.8·85-s − 1.62·97-s − 3.18·101-s − 2.68·109-s − 2.25·113-s − 9.30·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 10.6·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 5.53·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{32} \cdot 3^{8}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{2304} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{32} \cdot 3^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.870932961\)
\(L(\frac12)\) \(\approx\) \(1.870932961\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^3$ \( 1 - 206 T^{4} + p^{4} T^{8} \)
13$C_2^2$ \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
19$C_2^2$$\times$$C_2^2$ \( ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} ) \)
23$C_2^2$ \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 44 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \)
43$C_2^3$ \( 1 - 1198 T^{4} + p^{4} T^{8} \)
47$C_2^2$ \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^3$ \( 1 + 4946 T^{4} + p^{4} T^{8} \)
71$C_2^2$ \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \)
79$C_2^2$ \( ( 1 + 140 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^3$ \( 1 - 5678 T^{4} + p^{4} T^{8} \)
89$C_2^2$ \( ( 1 + 18 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
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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

−6.46787669523686087911171125593, −6.12037577768347044742358463764, −5.96017808759052343894584113349, −5.67748616431684756353462742180, −5.48051986940857899689547142481, −5.46575774805932600455139064870, −5.16784833704585551655689096009, −5.11198517483063446117853502887, −4.53037231747636614780756066953, −4.19038354001089997481501250416, −4.10765781821088163764500017615, −3.97848449908377327645617499458, −3.76489249449833151399478317322, −3.62421376643498998759795845427, −3.44187378014585571285971055976, −3.25648033433620113852056975679, −3.14274292787743549056078545070, −2.73742679783199083848775227706, −2.66509754649516254128699670644, −1.73884290339946242779963227575, −1.36442122673214005782188696933, −1.33475434791271000162803324306, −1.20732504073449967939306802557, −0.68095069863424648489079581465, −0.31732378595242006805976043831, 0.31732378595242006805976043831, 0.68095069863424648489079581465, 1.20732504073449967939306802557, 1.33475434791271000162803324306, 1.36442122673214005782188696933, 1.73884290339946242779963227575, 2.66509754649516254128699670644, 2.73742679783199083848775227706, 3.14274292787743549056078545070, 3.25648033433620113852056975679, 3.44187378014585571285971055976, 3.62421376643498998759795845427, 3.76489249449833151399478317322, 3.97848449908377327645617499458, 4.10765781821088163764500017615, 4.19038354001089997481501250416, 4.53037231747636614780756066953, 5.11198517483063446117853502887, 5.16784833704585551655689096009, 5.46575774805932600455139064870, 5.48051986940857899689547142481, 5.67748616431684756353462742180, 5.96017808759052343894584113349, 6.12037577768347044742358463764, 6.46787669523686087911171125593

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