↑

Properties

Label 8-1134e4-1.1-c3e4-0-3
Degree $8$
Conductor $1.654\times 10^{12}$
Sign $1$
Analytic cond. $2.00408\times 10^{7}$
Root an. cond. $8.17974$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 8·2-s + 40·4-s + 5-s − 28·7-s − 160·8-s − 8·10-s + 5·11-s − 21·13-s + 224·14-s + 560·16-s + 23·17-s − 94·19-s + 40·20-s − 40·22-s + 374·23-s − 229·25-s + 168·26-s − 1.12e3·28-s + 271·29-s − 243·31-s − 1.79e3·32-s − 184·34-s − 28·35-s − 181·37-s + 752·38-s − 160·40-s − 213·41-s + ⋯
L(s)  = 1  − 2.82·2-s + 5·4-s + 0.0894·5-s − 1.51·7-s − 7.07·8-s − 0.252·10-s + 0.137·11-s − 0.448·13-s + 4.27·14-s + 35/4·16-s + 0.328·17-s − 1.13·19-s + 0.447·20-s − 0.387·22-s + 3.39·23-s − 1.83·25-s + 1.26·26-s − 7.55·28-s + 1.73·29-s − 1.40·31-s − 9.89·32-s − 0.928·34-s − 0.135·35-s − 0.804·37-s + 3.21·38-s − 0.632·40-s − 0.811·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{16} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(2.00408\times 10^{7}\)
Root analytic conductor: \(8.17974\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{4} \cdot 3^{16} \cdot 7^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 \)
7$C_1$ \( ( 1 + p T )^{4} \)
good5$C_2 \wr S_4$ \( 1 - T + 46 p T^{2} - 82 p^{2} T^{3} + 24388 T^{4} - 82 p^{5} T^{5} + 46 p^{7} T^{6} - p^{9} T^{7} + p^{12} T^{8} \)
11$C_2 \wr S_4$ \( 1 - 5 T + 2207 T^{2} - 84743 T^{3} + 2019307 T^{4} - 84743 p^{3} T^{5} + 2207 p^{6} T^{6} - 5 p^{9} T^{7} + p^{12} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 21 T + 7045 T^{2} + 145926 T^{3} + 21382086 T^{4} + 145926 p^{3} T^{5} + 7045 p^{6} T^{6} + 21 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 23 T + 9893 T^{2} - 281573 T^{3} + 69040201 T^{4} - 281573 p^{3} T^{5} + 9893 p^{6} T^{6} - 23 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 94 T + 13819 T^{2} + 1292485 T^{3} + 131846581 T^{4} + 1292485 p^{3} T^{5} + 13819 p^{6} T^{6} + 94 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 374 T + 94043 T^{2} - 15913763 T^{3} + 2032574404 T^{4} - 15913763 p^{3} T^{5} + 94043 p^{6} T^{6} - 374 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 271 T + 112244 T^{2} - 18969550 T^{3} + 4222891402 T^{4} - 18969550 p^{3} T^{5} + 112244 p^{6} T^{6} - 271 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 243 T + 65200 T^{2} + 3464370 T^{3} + 918879468 T^{4} + 3464370 p^{3} T^{5} + 65200 p^{6} T^{6} + 243 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 181 T + 42844 T^{2} - 12321968 T^{3} - 3161946908 T^{4} - 12321968 p^{3} T^{5} + 42844 p^{6} T^{6} + 181 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 213 T + 129959 T^{2} + 46840167 T^{3} + 10098083799 T^{4} + 46840167 p^{3} T^{5} + 129959 p^{6} T^{6} + 213 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 238 T + 117319 T^{2} + 16596607 T^{3} + 11948476861 T^{4} + 16596607 p^{3} T^{5} + 117319 p^{6} T^{6} + 238 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 675 T + 418238 T^{2} - 179950518 T^{3} + 67529610954 T^{4} - 179950518 p^{3} T^{5} + 418238 p^{6} T^{6} - 675 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 54 T + 236591 T^{2} + 19653129 T^{3} + 33755250168 T^{4} + 19653129 p^{3} T^{5} + 236591 p^{6} T^{6} + 54 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 202 T + 571853 T^{2} - 66099259 T^{3} + 154363727461 T^{4} - 66099259 p^{3} T^{5} + 571853 p^{6} T^{6} - 202 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 1212 T + 1229677 T^{2} + 743602851 T^{3} + 422909187636 T^{4} + 743602851 p^{3} T^{5} + 1229677 p^{6} T^{6} + 1212 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 139 T + 819859 T^{2} - 196997395 T^{3} + 309834703945 T^{4} - 196997395 p^{3} T^{5} + 819859 p^{6} T^{6} - 139 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 1295 T + 1795244 T^{2} - 1349389658 T^{3} + 1011484858822 T^{4} - 1349389658 p^{3} T^{5} + 1795244 p^{6} T^{6} - 1295 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 2000 T + 2663509 T^{2} + 2429329973 T^{3} + 1727799097657 T^{4} + 2429329973 p^{3} T^{5} + 2663509 p^{6} T^{6} + 2000 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 1545 T + 2251384 T^{2} + 2027587356 T^{3} + 1673654255004 T^{4} + 2027587356 p^{3} T^{5} + 2251384 p^{6} T^{6} + 1545 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 142 T + 1002017 T^{2} - 646519193 T^{3} + 323485564444 T^{4} - 646519193 p^{3} T^{5} + 1002017 p^{6} T^{6} + 142 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 132 T + 738863 T^{2} + 1004820993 T^{3} - 79516549278 T^{4} + 1004820993 p^{3} T^{5} + 738863 p^{6} T^{6} - 132 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 638 T + 605011 T^{2} + 1006628117 T^{3} + 1509541105561 T^{4} + 1006628117 p^{3} T^{5} + 605011 p^{6} T^{6} + 638 p^{9} T^{7} + p^{12} T^{8} \)
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

−7.23264328213252893054420038500, −6.84312164896527278325980507033, −6.79114467234546307893328461155, −6.63272839261179719658455186334, −6.48147889122607613347065409613, −5.93166608615633506447358969741, −5.92559880569117062332248696049, −5.64437696330905866509547484028, −5.62898616251930145707141047394, −5.11493850143393814483210864679, −4.77871072715616601571725858230, −4.66825178678091214852160316965, −4.33572122024131238918247626305, −3.82296629803328674730338562111, −3.52093740680485179505074460930, −3.43252480170051474762280456574, −3.33403997112344410057940835745, −2.64255324795746228429653462000, −2.52723822246349682539449249070, −2.50494554357194344591104126975, −2.32123908776700091709990465741, −1.53668215942109502626040404842, −1.23749730710036417589695950708, −1.15960675393861845124767464417, −1.15567845780269637693841540198, 0, 0, 0, 0, 1.15567845780269637693841540198, 1.15960675393861845124767464417, 1.23749730710036417589695950708, 1.53668215942109502626040404842, 2.32123908776700091709990465741, 2.50494554357194344591104126975, 2.52723822246349682539449249070, 2.64255324795746228429653462000, 3.33403997112344410057940835745, 3.43252480170051474762280456574, 3.52093740680485179505074460930, 3.82296629803328674730338562111, 4.33572122024131238918247626305, 4.66825178678091214852160316965, 4.77871072715616601571725858230, 5.11493850143393814483210864679, 5.62898616251930145707141047394, 5.64437696330905866509547484028, 5.92559880569117062332248696049, 5.93166608615633506447358969741, 6.48147889122607613347065409613, 6.63272839261179719658455186334, 6.79114467234546307893328461155, 6.84312164896527278325980507033, 7.23264328213252893054420038500

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