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Properties

Label 8-2178e4-1.1-c3e4-0-2
Degree $8$
Conductor $2.250\times 10^{13}$
Sign $1$
Analytic cond. $2.72706\times 10^{8}$
Root an. cond. $11.3360$
Motivic weight $3$
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  + 8·2-s + 40·4-s + 6·5-s + 7-s + 160·8-s + 48·10-s − 18·13-s + 8·14-s + 560·16-s − 11·17-s − 75·19-s + 240·20-s + 58·23-s − 95·25-s − 144·26-s + 40·28-s − 245·29-s + 28·31-s + 1.79e3·32-s − 88·34-s + 6·35-s − 24·37-s − 600·38-s + 960·40-s + 512·41-s − 608·43-s + 464·46-s + ⋯
L(s)  = 1  + 2.82·2-s + 5·4-s + 0.536·5-s + 0.0539·7-s + 7.07·8-s + 1.51·10-s − 0.384·13-s + 0.152·14-s + 35/4·16-s − 0.156·17-s − 0.905·19-s + 2.68·20-s + 0.525·23-s − 0.759·25-s − 1.08·26-s + 0.269·28-s − 1.56·29-s + 0.162·31-s + 9.89·32-s − 0.443·34-s + 0.0289·35-s − 0.106·37-s − 2.56·38-s + 3.79·40-s + 1.95·41-s − 2.15·43-s + 1.48·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{8} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(2.72706\times 10^{8}\)
Root analytic conductor: \(11.3360\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{8} \cdot 11^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(33.57825765\)
\(L(\frac12)\) \(\approx\) \(33.57825765\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 - p T )^{4} \)
3 \( 1 \)
11 \( 1 \)
good5$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 131 T^{2} + 984 T^{3} + 1621 T^{4} + 984 p^{3} T^{5} + 131 p^{6} T^{6} - 6 p^{9} T^{7} + p^{12} T^{8} \)
7$C_2 \wr C_2\wr C_2$ \( 1 - T - 22 T^{2} + 5340 T^{3} + 81271 T^{4} + 5340 p^{3} T^{5} - 22 p^{6} T^{6} - p^{9} T^{7} + p^{12} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 18 T + 79 p T^{2} + 2220 p T^{3} + 8933536 T^{4} + 2220 p^{4} T^{5} + 79 p^{7} T^{6} + 18 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 + 11 T + 10203 T^{2} - 48435 T^{3} + 61656516 T^{4} - 48435 p^{3} T^{5} + 10203 p^{6} T^{6} + 11 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 75 T + 7031 T^{2} - 309675 T^{3} - 34250604 T^{4} - 309675 p^{3} T^{5} + 7031 p^{6} T^{6} + 75 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - 58 T + 20607 T^{2} - 1565100 T^{3} + 383374836 T^{4} - 1565100 p^{3} T^{5} + 20607 p^{6} T^{6} - 58 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 245 T + 89641 T^{2} + 15121415 T^{3} + 3298695156 T^{4} + 15121415 p^{3} T^{5} + 89641 p^{6} T^{6} + 245 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 - 28 T + 102453 T^{2} - 1509496 T^{3} + 4314653165 T^{4} - 1509496 p^{3} T^{5} + 102453 p^{6} T^{6} - 28 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 24 T + 101243 T^{2} - 779940 T^{3} + 6721018076 T^{4} - 779940 p^{3} T^{5} + 101243 p^{6} T^{6} + 24 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 512 T + 144193 T^{2} - 14849594 T^{3} + 2194275900 T^{4} - 14849594 p^{3} T^{5} + 144193 p^{6} T^{6} - 512 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 608 T + 279297 T^{2} + 90201630 T^{3} + 28742941456 T^{4} + 90201630 p^{3} T^{5} + 279297 p^{6} T^{6} + 608 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 129 T + 173893 T^{2} - 51438545 T^{3} + 20053053316 T^{4} - 51438545 p^{3} T^{5} + 173893 p^{6} T^{6} - 129 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 308 T + 375947 T^{2} - 1893730 p T^{3} + 81961053431 T^{4} - 1893730 p^{4} T^{5} + 375947 p^{6} T^{6} - 308 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 1030 T + 1073081 T^{2} + 629245210 T^{3} + 356065457891 T^{4} + 629245210 p^{3} T^{5} + 1073081 p^{6} T^{6} + 1030 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 343 T + 793503 T^{2} - 212359131 T^{3} + 261453595840 T^{4} - 212359131 p^{3} T^{5} + 793503 p^{6} T^{6} - 343 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 1611 T + 1721413 T^{2} - 1203436395 T^{3} + 737006756236 T^{4} - 1203436395 p^{3} T^{5} + 1721413 p^{6} T^{6} - 1611 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 573 T + 1259253 T^{2} + 487482501 T^{3} + 632135965840 T^{4} + 487482501 p^{3} T^{5} + 1259253 p^{6} T^{6} + 573 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 133 T + 1412097 T^{2} + 169268685 T^{3} + 796205426356 T^{4} + 169268685 p^{3} T^{5} + 1412097 p^{6} T^{6} + 133 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 1825 T + 3066836 T^{2} + 2912788200 T^{3} + 2536439736941 T^{4} + 2912788200 p^{3} T^{5} + 3066836 p^{6} T^{6} + 1825 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 287 T + 1246982 T^{2} + 80212670 T^{3} + 850973000681 T^{4} + 80212670 p^{3} T^{5} + 1246982 p^{6} T^{6} + 287 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 280 T + 2771741 T^{2} + 582852310 T^{3} + 2914840934036 T^{4} + 582852310 p^{3} T^{5} + 2771741 p^{6} T^{6} + 280 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 - 1816 T + 4101483 T^{2} - 4603173870 T^{3} + 5704133259811 T^{4} - 4603173870 p^{3} T^{5} + 4101483 p^{6} T^{6} - 1816 p^{9} T^{7} + p^{12} T^{8} \)
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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

−6.05504595338569605166232371428, −5.67974240696554825500104211797, −5.62951755790606306850396638899, −5.47424740155152966640758603766, −5.33598854805799893336101706464, −4.91599540787182216092116225581, −4.66952801523777578517189131147, −4.64489980098208795330079401262, −4.52526202526281170536554584361, −4.09997518562711841084094128737, −3.84827374113399892664908063718, −3.82515268513601346815808256495, −3.69992923130163080874339581969, −3.15282998625082048508447178868, −2.99429444469250214761749714195, −2.93767406019044919456309964520, −2.62743880756087755613574032050, −2.18387924290429949765595243032, −2.09998747964540584563864162812, −1.94705079666242902577160884325, −1.68492426482181202559113006362, −1.21861195771864549780294411469, −1.13619187656114463771161327087, −0.50110021756865756987353724169, −0.25665137310204882049149595500, 0.25665137310204882049149595500, 0.50110021756865756987353724169, 1.13619187656114463771161327087, 1.21861195771864549780294411469, 1.68492426482181202559113006362, 1.94705079666242902577160884325, 2.09998747964540584563864162812, 2.18387924290429949765595243032, 2.62743880756087755613574032050, 2.93767406019044919456309964520, 2.99429444469250214761749714195, 3.15282998625082048508447178868, 3.69992923130163080874339581969, 3.82515268513601346815808256495, 3.84827374113399892664908063718, 4.09997518562711841084094128737, 4.52526202526281170536554584361, 4.64489980098208795330079401262, 4.66952801523777578517189131147, 4.91599540787182216092116225581, 5.33598854805799893336101706464, 5.47424740155152966640758603766, 5.62951755790606306850396638899, 5.67974240696554825500104211797, 6.05504595338569605166232371428

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