Properties

Label 8-63e8-1.1-c1e4-0-1
Degree $8$
Conductor $2.482\times 10^{14}$
Sign $1$
Analytic cond. $1.00886\times 10^{6}$
Root an. cond. $5.62962$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s − 2·5-s − 3·8-s − 2·10-s + 5·11-s + 5·13-s − 4·16-s − 6·17-s + 8·19-s + 2·20-s + 5·22-s − 12·23-s − 4·25-s + 5·26-s − 10·29-s + 18·31-s − 32-s − 6·34-s + 8·38-s + 6·40-s + 5·41-s − 7·43-s − 5·44-s − 12·46-s + 21·47-s − 4·50-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 0.894·5-s − 1.06·8-s − 0.632·10-s + 1.50·11-s + 1.38·13-s − 16-s − 1.45·17-s + 1.83·19-s + 0.447·20-s + 1.06·22-s − 2.50·23-s − 4/5·25-s + 0.980·26-s − 1.85·29-s + 3.23·31-s − 0.176·32-s − 1.02·34-s + 1.29·38-s + 0.948·40-s + 0.780·41-s − 1.06·43-s − 0.753·44-s − 1.76·46-s + 3.06·47-s − 0.565·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{16} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(1.00886\times 10^{6}\)
Root analytic conductor: \(5.62962\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{16} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(5.931864108\)
\(L(\frac12)\) \(\approx\) \(5.931864108\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good2$C_2 \wr S_4$ \( 1 - T + p T^{2} + 3 T^{4} + p^{3} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
5$C_2 \wr S_4$ \( 1 + 2 T + 8 T^{2} - 3 T^{3} + 9 T^{4} - 3 p T^{5} + 8 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr S_4$ \( 1 - 5 T + 20 T^{2} + 9 T^{3} - 51 T^{4} + 9 p T^{5} + 20 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 5 T + 4 p T^{2} - 175 T^{3} + 1007 T^{4} - 175 p T^{5} + 4 p^{3} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 6 T + 23 T^{2} - 48 T^{3} - 363 T^{4} - 48 p T^{5} + 23 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr S_4$ \( 1 - 8 T + 4 p T^{2} - 409 T^{3} + 2117 T^{4} - 409 p T^{5} + 4 p^{3} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 12 T + 128 T^{2} + 861 T^{3} + 4839 T^{4} + 861 p T^{5} + 128 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 10 T + 4 p T^{2} + 780 T^{3} + 5109 T^{4} + 780 p T^{5} + 4 p^{3} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 18 T + 7 p T^{2} - 1810 T^{3} + 11511 T^{4} - 1810 p T^{5} + 7 p^{3} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 70 T^{2} + p T^{3} + 3393 T^{4} + p^{2} T^{5} + 70 p^{2} T^{6} + p^{4} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 5 T + 92 T^{2} - 489 T^{3} + 4623 T^{4} - 489 p T^{5} + 92 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 7 T + 142 T^{2} + 727 T^{3} + 8563 T^{4} + 727 p T^{5} + 142 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 21 T + 341 T^{2} - 3408 T^{3} + 28077 T^{4} - 3408 p T^{5} + 341 p^{2} T^{6} - 21 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 12 T + 4 p T^{2} + 1803 T^{3} + 16773 T^{4} + 1803 p T^{5} + 4 p^{3} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 6 T + 128 T^{2} + 861 T^{3} + 8331 T^{4} + 861 p T^{5} + 128 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 20 T + 271 T^{2} - 3010 T^{3} + 26663 T^{4} - 3010 p T^{5} + 271 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 5 T + 196 T^{2} + 931 T^{3} + 17639 T^{4} + 931 p T^{5} + 196 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 9 T + 257 T^{2} + 1782 T^{3} + 26655 T^{4} + 1782 p T^{5} + 257 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 6 T + 142 T^{2} - 19 p T^{3} + 12363 T^{4} - 19 p^{2} T^{5} + 142 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 10 T + 232 T^{2} + 2365 T^{3} + 24181 T^{4} + 2365 p T^{5} + 232 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 9 T + 206 T^{2} + 531 T^{3} + 15315 T^{4} + 531 p T^{5} + 206 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 22 T + 470 T^{2} - 5925 T^{3} + 67797 T^{4} - 5925 p T^{5} + 470 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 9 T + 130 T^{2} - 163 T^{3} + 9279 T^{4} - 163 p T^{5} + 130 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.03408994142541035718497942090, −5.96940305892795984735858616681, −5.59190863993422577663824187304, −5.43279298858584313380588055622, −5.17505190556480861272676832356, −4.74576498891906544153199171256, −4.66753495513223352876795549882, −4.61064064630552827239466530834, −4.39127546087758096615501225736, −3.99985338703509812609871819770, −3.90501477760028122168831666013, −3.87692461369736929243759681074, −3.78505056690742645761329061067, −3.30771754235213843518208389586, −3.21458999803352571794215077071, −3.08290729380123424840633994326, −2.51835798374604194388418362762, −2.47542209238087301684593740065, −2.16783341856723298124845887909, −1.70130598884824603243348196437, −1.69475257016936515482046706673, −1.41089077239383124672626750867, −0.66485794361622822959783680174, −0.56629340225449150365844002863, −0.53391333404621937392564518983, 0.53391333404621937392564518983, 0.56629340225449150365844002863, 0.66485794361622822959783680174, 1.41089077239383124672626750867, 1.69475257016936515482046706673, 1.70130598884824603243348196437, 2.16783341856723298124845887909, 2.47542209238087301684593740065, 2.51835798374604194388418362762, 3.08290729380123424840633994326, 3.21458999803352571794215077071, 3.30771754235213843518208389586, 3.78505056690742645761329061067, 3.87692461369736929243759681074, 3.90501477760028122168831666013, 3.99985338703509812609871819770, 4.39127546087758096615501225736, 4.61064064630552827239466530834, 4.66753495513223352876795549882, 4.74576498891906544153199171256, 5.17505190556480861272676832356, 5.43279298858584313380588055622, 5.59190863993422577663824187304, 5.96940305892795984735858616681, 6.03408994142541035718497942090

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