↑

Properties

Label 8-1250e4-1.1-c1e4-0-7
Degree $8$
Conductor $2.441\times 10^{12}$
Sign $1$
Analytic cond. $9925.41$
Root an. cond. $3.15931$
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  + 4·2-s + 3-s + 10·4-s + 4·6-s + 2·7-s + 20·8-s − 2·9-s − 2·11-s + 10·12-s + 11·13-s + 8·14-s + 35·16-s + 12·17-s − 8·18-s − 5·19-s + 2·21-s − 8·22-s − 4·23-s + 20·24-s + 44·26-s + 20·28-s + 15·29-s − 12·31-s + 56·32-s − 2·33-s + 48·34-s − 20·36-s + ⋯
L(s)  = 1  + 2.82·2-s + 0.577·3-s + 5·4-s + 1.63·6-s + 0.755·7-s + 7.07·8-s − 2/3·9-s − 0.603·11-s + 2.88·12-s + 3.05·13-s + 2.13·14-s + 35/4·16-s + 2.91·17-s − 1.88·18-s − 1.14·19-s + 0.436·21-s − 1.70·22-s − 0.834·23-s + 4.08·24-s + 8.62·26-s + 3.77·28-s + 2.78·29-s − 2.15·31-s + 9.89·32-s − 0.348·33-s + 8.23·34-s − 3.33·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 5^{16}\)
Sign: $1$
Analytic conductor: \(9925.41\)
Root analytic conductor: \(3.15931\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 5^{16} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(59.62574034\)
\(L(\frac12)\) \(\approx\) \(59.62574034\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{4} \)
5 \( 1 \)
good3$C_2^2:C_4$ \( 1 - T + p T^{2} - 5 T^{3} + 16 T^{4} - 5 p T^{5} + p^{3} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
7$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 17 T^{2} - 30 T^{3} + 156 T^{4} - 30 p T^{5} + 17 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 3 p T^{2} + 54 T^{3} + 500 T^{4} + 54 p T^{5} + 3 p^{3} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 - 11 T + 6 p T^{2} - 370 T^{3} + 1491 T^{4} - 370 p T^{5} + 6 p^{3} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 107 T^{2} - 610 T^{3} + 2951 T^{4} - 610 p T^{5} + 107 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2^2:C_4$ \( 1 + 5 T + 41 T^{2} + 265 T^{3} + 916 T^{4} + 265 p T^{5} + 41 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 63 T^{2} + 280 T^{3} + 1856 T^{4} + 280 p T^{5} + 63 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 - 15 T + 186 T^{2} - 1410 T^{3} + 9111 T^{4} - 1410 p T^{5} + 186 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 123 T^{2} + 684 T^{3} + 4440 T^{4} + 684 p T^{5} + 123 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 167 T^{2} - 1230 T^{3} + 9691 T^{4} - 1230 p T^{5} + 167 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 13 T + 198 T^{2} - 1576 T^{3} + 12785 T^{4} - 1576 p T^{5} + 198 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 133 T^{2} - 710 T^{3} + 7916 T^{4} - 710 p T^{5} + 133 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 177 T^{2} - 270 T^{3} + 12236 T^{4} - 270 p T^{5} + 177 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 11 T + 163 T^{2} - 1285 T^{3} + 11916 T^{4} - 1285 p T^{5} + 163 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 96 T^{2} - 560 T^{3} + 4046 T^{4} - 560 p T^{5} + 96 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 143 T^{2} - 506 T^{3} + 8295 T^{4} - 506 p T^{5} + 143 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 22 T + 337 T^{2} - 3850 T^{3} + 35236 T^{4} - 3850 p T^{5} + 337 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 22 T + 413 T^{2} + 4854 T^{3} + 48500 T^{4} + 4854 p T^{5} + 413 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - 21 T + 373 T^{2} - 4385 T^{3} + 42716 T^{4} - 4385 p T^{5} + 373 p^{2} T^{6} - 21 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 296 T^{2} + 2130 T^{3} + 33966 T^{4} + 2130 p T^{5} + 296 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 24 T + 463 T^{2} + 5520 T^{3} + 59416 T^{4} + 5520 p T^{5} + 463 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 5 T + 191 T^{2} - 165 T^{3} + 15056 T^{4} - 165 p T^{5} + 191 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
97$D_{4}$ \( ( 1 - 11 T + 193 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{2} \)
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

−6.78161164499013556983314773127, −6.45550755085176343693011772447, −6.24256390553439792661145275043, −6.15443569717554274599291572688, −6.02295408932589727708438262143, −5.74486543080365882251719944753, −5.62047103412040368908391583369, −5.35600553088722848839698008082, −5.21580460792579433225876224714, −4.91085236311740370166266310637, −4.60335335091713844743550512109, −4.36999709873338159325514500929, −4.15631402081250404976409429778, −3.87420591515630591006507556629, −3.68202699105073395621897230219, −3.48217081273564790397611396190, −3.46272383145165818613155055890, −2.88490860832851029813692273180, −2.63629409525392498914538667141, −2.55211905138264088552825177847, −2.38402247462264952478873044434, −1.76638449118684292516100565823, −1.38781423854849757152074640399, −1.03118147548760411110470121803, −1.00927063285389978481990054568, 1.00927063285389978481990054568, 1.03118147548760411110470121803, 1.38781423854849757152074640399, 1.76638449118684292516100565823, 2.38402247462264952478873044434, 2.55211905138264088552825177847, 2.63629409525392498914538667141, 2.88490860832851029813692273180, 3.46272383145165818613155055890, 3.48217081273564790397611396190, 3.68202699105073395621897230219, 3.87420591515630591006507556629, 4.15631402081250404976409429778, 4.36999709873338159325514500929, 4.60335335091713844743550512109, 4.91085236311740370166266310637, 5.21580460792579433225876224714, 5.35600553088722848839698008082, 5.62047103412040368908391583369, 5.74486543080365882251719944753, 6.02295408932589727708438262143, 6.15443569717554274599291572688, 6.24256390553439792661145275043, 6.45550755085176343693011772447, 6.78161164499013556983314773127

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