Properties

Label 8-180e4-1.1-c2e4-0-9
Degree $8$
Conductor $1049760000$
Sign $1$
Analytic cond. $578.669$
Root an. cond. $2.21464$
Motivic weight $2$
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  + 7·4-s + 33·16-s − 50·25-s − 196·49-s + 472·61-s + 119·64-s − 350·100-s − 88·109-s + 484·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 676·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 1.37e3·196-s + 197-s + 199-s + ⋯
L(s)  = 1  + 7/4·4-s + 2.06·16-s − 2·25-s − 4·49-s + 7.73·61-s + 1.85·64-s − 7/2·100-s − 0.807·109-s + 4·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 4·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s − 7·196-s + 0.00507·197-s + 0.00502·199-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 3^{8} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(578.669\)
Root analytic conductor: \(2.21464\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{180} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 3^{8} \cdot 5^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.926898889\)
\(L(\frac12)\) \(\approx\) \(3.926898889\)
\(L(2)\) 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$C_2^2$ \( 1 - 7 T^{2} + p^{4} T^{4} \)
3 \( 1 \)
5$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
good7$C_2$ \( ( 1 + p^{2} T^{2} )^{4} \)
11$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
13$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
17$C_2^2$ \( ( 1 - 382 T^{2} + p^{4} T^{4} )^{2} \)
19$C_2$ \( ( 1 - 22 T + p^{2} T^{2} )^{2}( 1 + 22 T + p^{2} T^{2} )^{2} \)
23$C_2^2$ \( ( 1 + 98 T^{2} + p^{4} T^{4} )^{2} \)
29$C_2$ \( ( 1 + p^{2} T^{2} )^{4} \)
31$C_2$ \( ( 1 - 2 T + p^{2} T^{2} )^{2}( 1 + 2 T + p^{2} T^{2} )^{2} \)
37$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
41$C_2$ \( ( 1 + p^{2} T^{2} )^{4} \)
43$C_2$ \( ( 1 + p^{2} T^{2} )^{4} \)
47$C_2^2$ \( ( 1 - 4222 T^{2} + p^{4} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + 1778 T^{2} + p^{4} T^{4} )^{2} \)
59$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
61$C_2$ \( ( 1 - 118 T + p^{2} T^{2} )^{4} \)
67$C_2$ \( ( 1 + p^{2} T^{2} )^{4} \)
71$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
73$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
79$C_2$ \( ( 1 - 98 T + p^{2} T^{2} )^{2}( 1 + 98 T + p^{2} T^{2} )^{2} \)
83$C_2^2$ \( ( 1 + 9938 T^{2} + p^{4} T^{4} )^{2} \)
89$C_2$ \( ( 1 + p^{2} T^{2} )^{4} \)
97$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
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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

−9.170748296611558363720919005926, −8.577556139829743073259236462164, −8.219653089388856729165542036522, −8.191772253089824194086351723127, −8.099139767922506522851436855834, −7.71511614337214403637906438029, −7.19612007305331832428950264862, −7.07039562035601777884115027117, −6.81560877430000993371432261209, −6.62451444504420816018397345711, −6.34282455026921302690943712313, −5.78794047065834910712782675606, −5.73481952696108318749166615691, −5.50147810588176706719404931623, −5.01489267642542554817514436976, −4.67387305211446898093836168124, −4.24590222009882136597292985208, −3.63575559246780927556807195483, −3.55505473700686498126572765941, −3.28431358399553533355316164406, −2.42700851535336932045025801689, −2.38899370000382688648787484083, −1.91892529519781189932154374780, −1.43094506493294664169337786972, −0.59616570877129900726359977756, 0.59616570877129900726359977756, 1.43094506493294664169337786972, 1.91892529519781189932154374780, 2.38899370000382688648787484083, 2.42700851535336932045025801689, 3.28431358399553533355316164406, 3.55505473700686498126572765941, 3.63575559246780927556807195483, 4.24590222009882136597292985208, 4.67387305211446898093836168124, 5.01489267642542554817514436976, 5.50147810588176706719404931623, 5.73481952696108318749166615691, 5.78794047065834910712782675606, 6.34282455026921302690943712313, 6.62451444504420816018397345711, 6.81560877430000993371432261209, 7.07039562035601777884115027117, 7.19612007305331832428950264862, 7.71511614337214403637906438029, 8.099139767922506522851436855834, 8.191772253089824194086351723127, 8.219653089388856729165542036522, 8.577556139829743073259236462164, 9.170748296611558363720919005926

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