Properties

Label 8-12e12-1.1-c1e4-0-28
Degree $8$
Conductor $8.916\times 10^{12}$
Sign $1$
Analytic cond. $36247.9$
Root an. cond. $3.71458$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 8·5-s + 6·7-s − 10·11-s − 10·13-s + 16·17-s + 6·19-s + 6·23-s + 44·25-s + 6·29-s + 8·31-s + 48·35-s + 12·37-s + 16·43-s − 2·47-s + 8·49-s + 16·53-s − 80·55-s + 6·59-s − 12·61-s − 80·65-s − 16·67-s − 60·77-s + 24·79-s − 2·83-s + 128·85-s − 60·91-s + 48·95-s + ⋯
L(s)  = 1  + 3.57·5-s + 2.26·7-s − 3.01·11-s − 2.77·13-s + 3.88·17-s + 1.37·19-s + 1.25·23-s + 44/5·25-s + 1.11·29-s + 1.43·31-s + 8.11·35-s + 1.97·37-s + 2.43·43-s − 0.291·47-s + 8/7·49-s + 2.19·53-s − 10.7·55-s + 0.781·59-s − 1.53·61-s − 9.92·65-s − 1.95·67-s − 6.83·77-s + 2.70·79-s − 0.219·83-s + 13.8·85-s − 6.28·91-s + 4.92·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 3^{12}\)
Sign: $1$
Analytic conductor: \(36247.9\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1728} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 3^{12} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(20.12118023\)
\(L(\frac12)\) \(\approx\) \(20.12118023\)
\(L(\frac{3}{2})\) 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$D_4\times C_2$ \( 1 - 8 T + 4 p T^{2} + 4 T^{3} - 89 T^{4} + 4 p T^{5} + 4 p^{3} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
7$D_4\times C_2$ \( 1 - 6 T + 4 p T^{2} - 96 T^{3} + 291 T^{4} - 96 p T^{5} + 4 p^{3} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
11$D_4\times C_2$ \( 1 + 10 T + 41 T^{2} + 82 T^{3} + 136 T^{4} + 82 p T^{5} + 41 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2$$\times$$C_2^2$ \( ( 1 + 5 T + p T^{2} )^{2}( 1 + 23 T^{2} + p^{2} T^{4} ) \)
17$C_2^2$ \( ( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 - 6 T + 18 T^{2} - 132 T^{3} + 959 T^{4} - 132 p T^{5} + 18 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
23$D_4\times C_2$ \( 1 - 6 T + 52 T^{2} - 240 T^{3} + 1347 T^{4} - 240 p T^{5} + 52 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2^3$ \( 1 - 6 T + 18 T^{2} - 36 T^{3} - 457 T^{4} - 36 p T^{5} + 18 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
31$D_4\times C_2$ \( 1 - 8 T - 2 T^{2} - 32 T^{3} + 1411 T^{4} - 32 p T^{5} - 2 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
37$D_4\times C_2$ \( 1 - 12 T + 72 T^{2} - 588 T^{3} + 4658 T^{4} - 588 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2^3$ \( 1 + 73 T^{2} + 3648 T^{4} + 73 p^{2} T^{6} + p^{4} T^{8} \)
43$D_4\times C_2$ \( 1 - 16 T + 65 T^{2} + 624 T^{3} - 8092 T^{4} + 624 p T^{5} + 65 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 + 2 T - 16 T^{2} - 148 T^{3} - 1997 T^{4} - 148 p T^{5} - 16 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 - 16 T + 128 T^{2} - 976 T^{3} + 7378 T^{4} - 976 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
59$D_4\times C_2$ \( 1 - 6 T + 45 T^{2} + 594 T^{3} - 3376 T^{4} + 594 p T^{5} + 45 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 + 12 T + 180 T^{2} + 1596 T^{3} + 15143 T^{4} + 1596 p T^{5} + 180 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 + 16 T + 113 T^{2} + 384 T^{3} - 172 T^{4} + 384 p T^{5} + 113 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
71$D_4\times C_2$ \( 1 - 156 T^{2} + 13094 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8} \)
73$D_4\times C_2$ \( 1 - 158 T^{2} + 16131 T^{4} - 158 p^{2} T^{6} + p^{4} T^{8} \)
79$C_2^2$ \( ( 1 - 12 T + 65 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2^3$ \( 1 + 2 T + 2 T^{2} - 328 T^{3} - 7217 T^{4} - 328 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2^2$ \( ( 1 - 174 T^{2} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 + 20 T + 109 T^{2} + 20 p T^{3} + 376 p T^{4} + 20 p^{2} T^{5} + 109 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.74392525762577163173916360405, −6.26058785291757868369048916566, −5.94502945978198527284105974404, −5.82278684955675445753518697344, −5.79299651406834563412061220869, −5.45051269111685758408101942678, −5.29149361343509990926232856118, −5.26038041375539002259844229219, −5.01393902384560110818772415562, −4.82157672420093057062033207925, −4.63684809192438005860293216043, −4.49574394085364304428198265458, −4.45945386841373724524389635188, −3.26830725285268289836609606098, −3.23911703973981658154019301280, −3.13671282698050593334454259811, −2.95694359390095642655836284027, −2.39243781198595182523525984267, −2.34044043373154937088283611177, −2.31410539464191925640222046093, −2.15028089707263274171209692966, −1.44186056819739029083951029326, −0.981273234775047120672720216413, −0.974007487622345031718137589208, −0.972848003299635026864511735236, 0.972848003299635026864511735236, 0.974007487622345031718137589208, 0.981273234775047120672720216413, 1.44186056819739029083951029326, 2.15028089707263274171209692966, 2.31410539464191925640222046093, 2.34044043373154937088283611177, 2.39243781198595182523525984267, 2.95694359390095642655836284027, 3.13671282698050593334454259811, 3.23911703973981658154019301280, 3.26830725285268289836609606098, 4.45945386841373724524389635188, 4.49574394085364304428198265458, 4.63684809192438005860293216043, 4.82157672420093057062033207925, 5.01393902384560110818772415562, 5.26038041375539002259844229219, 5.29149361343509990926232856118, 5.45051269111685758408101942678, 5.79299651406834563412061220869, 5.82278684955675445753518697344, 5.94502945978198527284105974404, 6.26058785291757868369048916566, 6.74392525762577163173916360405

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