↑

Properties

Label 8-1305e4-1.1-c1e4-0-9
Degree $8$
Conductor $2.900\times 10^{12}$
Sign $1$
Analytic cond. $11790.9$
Root an. cond. $3.22807$
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  + 3·2-s + 3·4-s + 4·5-s + 2·7-s + 2·8-s + 12·10-s + 2·11-s − 8·13-s + 6·14-s + 4·16-s + 10·17-s − 2·19-s + 12·20-s + 6·22-s + 12·23-s + 10·25-s − 24·26-s + 6·28-s + 4·29-s − 4·31-s + 4·32-s + 30·34-s + 8·35-s − 16·37-s − 6·38-s + 8·40-s + 12·41-s + ⋯
L(s)  = 1  + 2.12·2-s + 3/2·4-s + 1.78·5-s + 0.755·7-s + 0.707·8-s + 3.79·10-s + 0.603·11-s − 2.21·13-s + 1.60·14-s + 16-s + 2.42·17-s − 0.458·19-s + 2.68·20-s + 1.27·22-s + 2.50·23-s + 2·25-s − 4.70·26-s + 1.13·28-s + 0.742·29-s − 0.718·31-s + 0.707·32-s + 5.14·34-s + 1.35·35-s − 2.63·37-s − 0.973·38-s + 1.26·40-s + 1.87·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{8} \cdot 5^{4} \cdot 29^{4}\)
Sign: $1$
Analytic conductor: \(11790.9\)
Root analytic conductor: \(3.22807\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{8} \cdot 5^{4} \cdot 29^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(24.20399522\)
\(L(\frac12)\) \(\approx\) \(24.20399522\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
5$C_1$ \( ( 1 - T )^{4} \)
29$C_1$ \( ( 1 - T )^{4} \)
good2$C_4\wr C_2$ \( 1 - 3 T + 3 p T^{2} - 11 T^{3} + 17 T^{4} - 11 p T^{5} + 3 p^{3} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
7$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 13 T^{2} - 46 T^{3} + 88 T^{4} - 46 p T^{5} + 13 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 29 T^{2} - 70 T^{3} + 400 T^{4} - 70 p T^{5} + 29 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 55 T^{2} + 220 T^{3} + 928 T^{4} + 220 p T^{5} + 55 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 89 T^{2} - 470 T^{3} + 2332 T^{4} - 470 p T^{5} + 89 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 36 T^{2} + 58 T^{3} + 630 T^{4} + 58 p T^{5} + 36 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 80 T^{2} - 380 T^{3} + 1598 T^{4} - 380 p T^{5} + 80 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 64 T^{2} + 404 T^{3} + 2110 T^{4} + 404 p T^{5} + 64 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 160 T^{2} + 1040 T^{3} + 6478 T^{4} + 1040 p T^{5} + 160 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 84 T^{2} - 340 T^{3} + 1910 T^{4} - 340 p T^{5} + 84 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 56 T^{2} - 66 T^{3} + 2334 T^{4} - 66 p T^{5} + 56 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 187 T^{2} - 1620 T^{3} + 13096 T^{4} - 1620 p T^{5} + 187 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 132 T^{2} - 990 T^{3} + 8774 T^{4} - 990 p T^{5} + 132 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 24 T^{2} + 226 T^{3} + 6366 T^{4} + 226 p T^{5} + 24 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 26 T + 316 T^{2} + 2622 T^{3} + 19766 T^{4} + 2622 p T^{5} + 316 p^{2} T^{6} + 26 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 161 T^{2} - 694 T^{3} + 12472 T^{4} - 694 p T^{5} + 161 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 240 T^{2} - 1810 T^{3} + 24062 T^{4} - 1810 p T^{5} + 240 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 208 T^{2} + 320 T^{3} + 19454 T^{4} + 320 p T^{5} + 208 p^{2} T^{6} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 22 T + 324 T^{2} - 3926 T^{3} + 41126 T^{4} - 3926 p T^{5} + 324 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 196 T^{2} - 2690 T^{3} + 18742 T^{4} - 2690 p T^{5} + 196 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 67 T^{2} - 32 T^{3} + 6240 T^{4} - 32 p T^{5} + 67 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 22 T + 348 T^{2} + 4586 T^{3} + 51078 T^{4} + 4586 p T^{5} + 348 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \)
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

−6.88022676623477708125217688283, −6.61352613631778993665096036846, −6.26802939850415108535951581094, −6.07866951918344313202125584355, −5.93146675114601501052917610373, −5.48787434677885647395306902480, −5.41505195687278823753276083115, −5.34064961383050596117241062739, −5.16015307118862030253278861309, −4.71050638380219598717202588461, −4.70777471881903065136635914776, −4.65492507208300376837167956171, −4.46522086381216913223976409713, −3.82460131639147294173088142176, −3.79887005819955356746062876026, −3.29358622387056447369279713306, −3.25606953653899693689065639521, −3.08369922037493253462869783834, −2.59164670677942975261980067788, −2.26388666047203924476833680422, −2.11165322967482828365917688400, −1.84436193522081983643835915560, −1.26988037798009943994208634503, −1.11887033704630432353220121241, −0.67013277183930208911141896232, 0.67013277183930208911141896232, 1.11887033704630432353220121241, 1.26988037798009943994208634503, 1.84436193522081983643835915560, 2.11165322967482828365917688400, 2.26388666047203924476833680422, 2.59164670677942975261980067788, 3.08369922037493253462869783834, 3.25606953653899693689065639521, 3.29358622387056447369279713306, 3.79887005819955356746062876026, 3.82460131639147294173088142176, 4.46522086381216913223976409713, 4.65492507208300376837167956171, 4.70777471881903065136635914776, 4.71050638380219598717202588461, 5.16015307118862030253278861309, 5.34064961383050596117241062739, 5.41505195687278823753276083115, 5.48787434677885647395306902480, 5.93146675114601501052917610373, 6.07866951918344313202125584355, 6.26802939850415108535951581094, 6.61352613631778993665096036846, 6.88022676623477708125217688283

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