Properties

Label 8-6864e4-1.1-c1e4-0-6
Degree $8$
Conductor $2.220\times 10^{15}$
Sign $1$
Analytic cond. $9.02438\times 10^{6}$
Root an. cond. $7.40333$
Motivic weight $1$
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  + 4·3-s − 3·5-s − 3·7-s + 10·9-s + 4·11-s − 4·13-s − 12·15-s − 2·19-s − 12·21-s − 3·23-s − 3·25-s + 20·27-s − 21·29-s − 10·31-s + 16·33-s + 9·35-s + 4·37-s − 16·39-s − 15·41-s + 3·43-s − 30·45-s − 4·47-s − 7·49-s − 8·53-s − 12·55-s − 8·57-s + 59-s + ⋯
L(s)  = 1  + 2.30·3-s − 1.34·5-s − 1.13·7-s + 10/3·9-s + 1.20·11-s − 1.10·13-s − 3.09·15-s − 0.458·19-s − 2.61·21-s − 0.625·23-s − 3/5·25-s + 3.84·27-s − 3.89·29-s − 1.79·31-s + 2.78·33-s + 1.52·35-s + 0.657·37-s − 2.56·39-s − 2.34·41-s + 0.457·43-s − 4.47·45-s − 0.583·47-s − 49-s − 1.09·53-s − 1.61·55-s − 1.05·57-s + 0.130·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 11^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 11^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{4} \cdot 11^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(9.02438\times 10^{6}\)
Root analytic conductor: \(7.40333\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 11^{4} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_1$ \( ( 1 - T )^{4} \)
11$C_1$ \( ( 1 - T )^{4} \)
13$C_1$ \( ( 1 + T )^{4} \)
good5$C_2 \wr S_4$ \( 1 + 3 T + 12 T^{2} + 33 T^{3} + 18 p T^{4} + 33 p T^{5} + 12 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
7$C_2 \wr S_4$ \( 1 + 3 T + 16 T^{2} + 23 T^{3} + 110 T^{4} + 23 p T^{5} + 16 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 40 T^{2} - 36 T^{3} + 798 T^{4} - 36 p T^{5} + 40 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 2 T + 52 T^{2} + 66 T^{3} + 1286 T^{4} + 66 p T^{5} + 52 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 3 T - 4 T^{2} + 103 T^{3} + 854 T^{4} + 103 p T^{5} - 4 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 21 T + 228 T^{2} + 1755 T^{3} + 10602 T^{4} + 1755 p T^{5} + 228 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 10 T + 80 T^{2} + 462 T^{3} + 2382 T^{4} + 462 p T^{5} + 80 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 4 T + 52 T^{2} + 180 T^{3} + 134 T^{4} + 180 p T^{5} + 52 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 15 T + 190 T^{2} + 1401 T^{3} + 258 p T^{4} + 1401 p T^{5} + 190 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 3 T + 126 T^{2} - 423 T^{3} + 7134 T^{4} - 423 p T^{5} + 126 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 4 T + 92 T^{2} + 180 T^{3} + 4742 T^{4} + 180 p T^{5} + 92 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
59$C_2 \wr S_4$ \( 1 - T + 172 T^{2} - 161 T^{3} + 14134 T^{4} - 161 p T^{5} + 172 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 9 T + 118 T^{2} + 891 T^{3} + 10194 T^{4} + 891 p T^{5} + 118 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 5 T + 190 T^{2} - 669 T^{3} + 17246 T^{4} - 669 p T^{5} + 190 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 18 T + 276 T^{2} + 2650 T^{3} + 26102 T^{4} + 2650 p T^{5} + 276 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 27 T + 502 T^{2} + 6301 T^{3} + 62594 T^{4} + 6301 p T^{5} + 502 p^{2} T^{6} + 27 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 8 T + 312 T^{2} + 1780 T^{3} + 36662 T^{4} + 1780 p T^{5} + 312 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 14 T + 148 T^{2} - 1726 T^{3} + 20134 T^{4} - 1726 p T^{5} + 148 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 20 T + 396 T^{2} + 4616 T^{3} + 53942 T^{4} + 4616 p T^{5} + 396 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 4 T + 292 T^{2} - 540 T^{3} + 36854 T^{4} - 540 p T^{5} + 292 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
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

−6.08297825431626703958083338636, −5.77809062200774528885083415575, −5.48650876704672013169035408176, −5.43997536194154430873741304314, −5.35068690952942445993184406929, −4.86782594902414019777558538401, −4.64101113081238359297438878641, −4.52038140080170433035025325220, −4.41134592296333380276876595113, −3.96743154727291190908728383430, −3.93496227636032579774937258761, −3.89621258821204169436883507538, −3.78877355899874920927129861653, −3.38500094392845237217112100946, −3.23330789308265375217363132484, −3.14360197769000286279902462798, −3.08566134295612453109548105548, −2.73178130994908457143786968496, −2.32835233637072389998517566701, −2.13662451017918548529333565682, −2.09834180615357068734978167867, −1.62605741387409992900314508734, −1.52950115257319226873209170406, −1.51310498586409503511353922514, −1.06214217381596443893737668148, 0, 0, 0, 0, 1.06214217381596443893737668148, 1.51310498586409503511353922514, 1.52950115257319226873209170406, 1.62605741387409992900314508734, 2.09834180615357068734978167867, 2.13662451017918548529333565682, 2.32835233637072389998517566701, 2.73178130994908457143786968496, 3.08566134295612453109548105548, 3.14360197769000286279902462798, 3.23330789308265375217363132484, 3.38500094392845237217112100946, 3.78877355899874920927129861653, 3.89621258821204169436883507538, 3.93496227636032579774937258761, 3.96743154727291190908728383430, 4.41134592296333380276876595113, 4.52038140080170433035025325220, 4.64101113081238359297438878641, 4.86782594902414019777558538401, 5.35068690952942445993184406929, 5.43997536194154430873741304314, 5.48650876704672013169035408176, 5.77809062200774528885083415575, 6.08297825431626703958083338636

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