Properties

Label 8-882e4-1.1-c2e4-0-5
Degree $8$
Conductor $605165749776$
Sign $1$
Analytic cond. $333591.$
Root an. cond. $4.90232$
Motivic weight $2$
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  − 2·4-s + 12·5-s − 12·11-s − 12·17-s − 12·19-s − 24·20-s + 12·23-s + 40·25-s − 120·29-s + 72·31-s + 40·37-s − 128·43-s + 24·44-s − 168·47-s − 108·53-s − 144·55-s − 168·59-s + 24·61-s + 8·64-s − 88·67-s + 24·68-s + 120·71-s − 24·73-s + 24·76-s − 64·79-s − 144·85-s + 324·89-s + ⋯
L(s)  = 1  − 1/2·4-s + 12/5·5-s − 1.09·11-s − 0.705·17-s − 0.631·19-s − 6/5·20-s + 0.521·23-s + 8/5·25-s − 4.13·29-s + 2.32·31-s + 1.08·37-s − 2.97·43-s + 6/11·44-s − 3.57·47-s − 2.03·53-s − 2.61·55-s − 2.84·59-s + 0.393·61-s + 1/8·64-s − 1.31·67-s + 6/17·68-s + 1.69·71-s − 0.328·73-s + 6/19·76-s − 0.810·79-s − 1.69·85-s + 3.64·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(333591.\)
Root analytic conductor: \(4.90232\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{882} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{8} \cdot 7^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8422414485\)
\(L(\frac12)\) \(\approx\) \(0.8422414485\)
\(L(2)\) 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$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
3 \( 1 \)
7 \( 1 \)
good5$D_4\times C_2$ \( 1 - 12 T + 104 T^{2} - 672 T^{3} + 3711 T^{4} - 672 p^{2} T^{5} + 104 p^{4} T^{6} - 12 p^{6} T^{7} + p^{8} T^{8} \)
11$D_4\times C_2$ \( 1 + 12 T - 116 T^{2} + 216 T^{3} + 35535 T^{4} + 216 p^{2} T^{5} - 116 p^{4} T^{6} + 12 p^{6} T^{7} + p^{8} T^{8} \)
13$D_4\times C_2$ \( 1 - 244 T^{2} + 53574 T^{4} - 244 p^{4} T^{6} + p^{8} T^{8} \)
17$C_2^3$ \( 1 + 12 T - 88 T^{2} - 96 p T^{3} - 177 p^{2} T^{4} - 96 p^{3} T^{5} - 88 p^{4} T^{6} + 12 p^{6} T^{7} + p^{8} T^{8} \)
19$D_4\times C_2$ \( 1 + 12 T + 398 T^{2} + 4200 T^{3} + 9507 T^{4} + 4200 p^{2} T^{5} + 398 p^{4} T^{6} + 12 p^{6} T^{7} + p^{8} T^{8} \)
23$D_4\times C_2$ \( 1 - 12 T - 788 T^{2} + 1512 T^{3} + 512607 T^{4} + 1512 p^{2} T^{5} - 788 p^{4} T^{6} - 12 p^{6} T^{7} + p^{8} T^{8} \)
29$C_2$ \( ( 1 + 30 T + p^{2} T^{2} )^{4} \)
31$D_4\times C_2$ \( 1 - 72 T + 3218 T^{2} - 107280 T^{3} + 2957187 T^{4} - 107280 p^{2} T^{5} + 3218 p^{4} T^{6} - 72 p^{6} T^{7} + p^{8} T^{8} \)
37$D_4\times C_2$ \( 1 - 40 T + 1054 T^{2} + 87680 T^{3} - 3766445 T^{4} + 87680 p^{2} T^{5} + 1054 p^{4} T^{6} - 40 p^{6} T^{7} + p^{8} T^{8} \)
41$D_4\times C_2$ \( 1 - 4960 T^{2} + 11110434 T^{4} - 4960 p^{4} T^{6} + p^{8} T^{8} \)
43$D_{4}$ \( ( 1 + 64 T + 2922 T^{2} + 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 + 168 T + 16082 T^{2} + 23856 p T^{3} + 27363 p^{2} T^{4} + 23856 p^{3} T^{5} + 16082 p^{4} T^{6} + 168 p^{6} T^{7} + p^{8} T^{8} \)
53$D_4\times C_2$ \( 1 + 108 T + 3418 T^{2} + 283824 T^{3} + 27341859 T^{4} + 283824 p^{2} T^{5} + 3418 p^{4} T^{6} + 108 p^{6} T^{7} + p^{8} T^{8} \)
59$D_4\times C_2$ \( 1 + 168 T + 16322 T^{2} + 1161552 T^{3} + 68435283 T^{4} + 1161552 p^{2} T^{5} + 16322 p^{4} T^{6} + 168 p^{6} T^{7} + p^{8} T^{8} \)
61$D_4\times C_2$ \( 1 - 24 T + 7586 T^{2} - 177456 T^{3} + 41539827 T^{4} - 177456 p^{2} T^{5} + 7586 p^{4} T^{6} - 24 p^{6} T^{7} + p^{8} T^{8} \)
67$D_4\times C_2$ \( 1 + 88 T - 2882 T^{2} + 145024 T^{3} + 57997939 T^{4} + 145024 p^{2} T^{5} - 2882 p^{4} T^{6} + 88 p^{6} T^{7} + p^{8} T^{8} \)
71$D_{4}$ \( ( 1 - 60 T + 4484 T^{2} - 60 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + 24 T + 6842 T^{2} + 159600 T^{3} + 16847427 T^{4} + 159600 p^{2} T^{5} + 6842 p^{4} T^{6} + 24 p^{6} T^{7} + p^{8} T^{8} \)
79$D_4\times C_2$ \( 1 + 64 T + 958 T^{2} - 598016 T^{3} - 54666173 T^{4} - 598016 p^{2} T^{5} + 958 p^{4} T^{6} + 64 p^{6} T^{7} + p^{8} T^{8} \)
83$D_4\times C_2$ \( 1 - 16612 T^{2} + 134415078 T^{4} - 16612 p^{4} T^{6} + p^{8} T^{8} \)
89$D_4\times C_2$ \( 1 - 324 T + 56936 T^{2} - 7109856 T^{3} + 695968527 T^{4} - 7109856 p^{2} T^{5} + 56936 p^{4} T^{6} - 324 p^{6} T^{7} + p^{8} T^{8} \)
97$D_4\times C_2$ \( 1 - 24820 T^{2} + 317127462 T^{4} - 24820 p^{4} T^{6} + p^{8} T^{8} \)
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

−7.03081891591510380205932094030, −6.62081412992006639366443538463, −6.46640203958892971100473053435, −6.33098478832578716008250703916, −6.18238746436281134847069551635, −5.87875681650538460872386580260, −5.73515898175766480078559266322, −5.61937587440355028132122202181, −4.98353803241610461310019646284, −4.95186878849856662353337931097, −4.86482105997887005887046534551, −4.56290672041748406329784216334, −4.52260477335677858684835671446, −3.84792012179766223479118128536, −3.53808101983755560309554913366, −3.28203970352737067231800353767, −3.13593102846698446465642751329, −2.90144904787136784538519416486, −2.22549953683339364032242070869, −2.04423729808501822801086194257, −1.90057490459076105023887848499, −1.75324537830936102122068751571, −1.39075300145093798996549810025, −0.60276316800708090908506125115, −0.15526586387125165489477626128, 0.15526586387125165489477626128, 0.60276316800708090908506125115, 1.39075300145093798996549810025, 1.75324537830936102122068751571, 1.90057490459076105023887848499, 2.04423729808501822801086194257, 2.22549953683339364032242070869, 2.90144904787136784538519416486, 3.13593102846698446465642751329, 3.28203970352737067231800353767, 3.53808101983755560309554913366, 3.84792012179766223479118128536, 4.52260477335677858684835671446, 4.56290672041748406329784216334, 4.86482105997887005887046534551, 4.95186878849856662353337931097, 4.98353803241610461310019646284, 5.61937587440355028132122202181, 5.73515898175766480078559266322, 5.87875681650538460872386580260, 6.18238746436281134847069551635, 6.33098478832578716008250703916, 6.46640203958892971100473053435, 6.62081412992006639366443538463, 7.03081891591510380205932094030

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