↑

Properties

Label 8-44e8-1.1-c3e4-0-5
Degree $8$
Conductor $1.405\times 10^{13}$
Sign $1$
Analytic cond. $1.70249\times 10^{8}$
Root an. cond. $10.6877$
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  − 4·3-s + 25·5-s + 3·7-s + 5·9-s + 41·13-s − 100·15-s − 52·17-s + 16·19-s − 12·21-s − 314·23-s + 52·25-s − 58·27-s − 561·29-s − 199·31-s + 75·35-s + 357·37-s − 164·39-s − 32·41-s − 721·43-s + 125·45-s − 403·47-s − 270·49-s + 208·51-s − 133·53-s − 64·57-s − 1.01e3·59-s + 919·61-s + ⋯
L(s)  = 1  − 0.769·3-s + 2.23·5-s + 0.161·7-s + 5/27·9-s + 0.874·13-s − 1.72·15-s − 0.741·17-s + 0.193·19-s − 0.124·21-s − 2.84·23-s + 0.415·25-s − 0.413·27-s − 3.59·29-s − 1.15·31-s + 0.362·35-s + 1.58·37-s − 0.673·39-s − 0.121·41-s − 2.55·43-s + 0.414·45-s − 1.25·47-s − 0.787·49-s + 0.571·51-s − 0.344·53-s − 0.148·57-s − 2.24·59-s + 1.92·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(1.70249\times 10^{8}\)
Root analytic conductor: \(10.6877\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{16} \cdot 11^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

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 \( 1 \)
11 \( 1 \)
good3$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 11 T^{2} + 82 T^{3} + 1315 T^{4} + 82 p^{3} T^{5} + 11 p^{6} T^{6} + 4 p^{9} T^{7} + p^{12} T^{8} \)
5$C_2 \wr C_2\wr C_2$ \( 1 - p^{2} T + 573 T^{2} - 1533 p T^{3} + 103716 T^{4} - 1533 p^{4} T^{5} + 573 p^{6} T^{6} - p^{11} T^{7} + p^{12} T^{8} \)
7$C_2 \wr C_2\wr C_2$ \( 1 - 3 T + 279 T^{2} - 1591 T^{3} + 251100 T^{4} - 1591 p^{3} T^{5} + 279 p^{6} T^{6} - 3 p^{9} T^{7} + p^{12} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 - 41 T + 5637 T^{2} - 241261 T^{3} + 15241940 T^{4} - 241261 p^{3} T^{5} + 5637 p^{6} T^{6} - 41 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 + 52 T + 16573 T^{2} + 591848 T^{3} + 112551705 T^{4} + 591848 p^{3} T^{5} + 16573 p^{6} T^{6} + 52 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 14825 T^{2} - 500832 T^{3} + 111187933 T^{4} - 500832 p^{3} T^{5} + 14825 p^{6} T^{6} - 16 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 + 314 T + 61816 T^{2} + 8042722 T^{3} + 950275950 T^{4} + 8042722 p^{3} T^{5} + 61816 p^{6} T^{6} + 314 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 561 T + 200635 T^{2} + 47945667 T^{3} + 8699420208 T^{4} + 47945667 p^{3} T^{5} + 200635 p^{6} T^{6} + 561 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 199 T + 51813 T^{2} - 256853 T^{3} + 389222024 T^{4} - 256853 p^{3} T^{5} + 51813 p^{6} T^{6} + 199 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 - 357 T + 160529 T^{2} - 42781809 T^{3} + 12141822140 T^{4} - 42781809 p^{3} T^{5} + 160529 p^{6} T^{6} - 357 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 32 T + 194093 T^{2} + 3202364 T^{3} + 18031305245 T^{4} + 3202364 p^{3} T^{5} + 194093 p^{6} T^{6} + 32 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 721 T + 420117 T^{2} + 154459221 T^{3} + 51447883420 T^{4} + 154459221 p^{3} T^{5} + 420117 p^{6} T^{6} + 721 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 403 T + 357463 T^{2} + 121226987 T^{3} + 52982120160 T^{4} + 121226987 p^{3} T^{5} + 357463 p^{6} T^{6} + 403 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 133 T + 365887 T^{2} + 13770175 T^{3} + 65244311076 T^{4} + 13770175 p^{3} T^{5} + 365887 p^{6} T^{6} + 133 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 1016 T + 1078585 T^{2} + 639078152 T^{3} + 358860597453 T^{4} + 639078152 p^{3} T^{5} + 1078585 p^{6} T^{6} + 1016 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 919 T + 1093783 T^{2} - 606559837 T^{3} + 388669851044 T^{4} - 606559837 p^{3} T^{5} + 1093783 p^{6} T^{6} - 919 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 289 T + 686735 T^{2} + 243308709 T^{3} + 267199450216 T^{4} + 243308709 p^{3} T^{5} + 686735 p^{6} T^{6} + 289 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 1205 T + 1608131 T^{2} - 1210556495 T^{3} + 899763338876 T^{4} - 1210556495 p^{3} T^{5} + 1608131 p^{6} T^{6} - 1205 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - 1234 T + 1514091 T^{2} - 1290056552 T^{3} + 893250631145 T^{4} - 1290056552 p^{3} T^{5} + 1514091 p^{6} T^{6} - 1234 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 603 T + 1688657 T^{2} + 808010451 T^{3} + 1205184633704 T^{4} + 808010451 p^{3} T^{5} + 1688657 p^{6} T^{6} + 603 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 1514 T + 2070047 T^{2} - 2046213464 T^{3} + 1753896205685 T^{4} - 2046213464 p^{3} T^{5} + 2070047 p^{6} T^{6} - 1514 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 1101 T + 2406895 T^{2} + 1914547747 T^{3} + 2334666485008 T^{4} + 1914547747 p^{3} T^{5} + 2406895 p^{6} T^{6} + 1101 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 - 2116 T + 3632073 T^{2} - 4516910940 T^{3} + 5180506457941 T^{4} - 4516910940 p^{3} T^{5} + 3632073 p^{6} T^{6} - 2116 p^{9} T^{7} + p^{12} T^{8} \)
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

−6.46837278826693760145734311562, −6.21849682748122146879098170753, −6.19773151345474869146988092202, −5.91277781717168167058725334013, −5.89245882144950373108338671439, −5.46791064519424492749877204619, −5.37413085475493896682548107564, −5.27786558896493820709254378740, −5.16677825462076977193601717194, −4.67341468752677984372186100092, −4.50115493962109934537005616677, −4.15070629559026359035880765486, −4.03708258876548845987557570353, −3.69208106538011908116223916334, −3.48002428696893587358138312408, −3.44467801549672841034346021419, −3.21180789664283586279885843703, −2.36184349748366255537354965017, −2.32672690624002495080543122039, −2.22285611311248012039573578777, −1.93204854835355817059576715164, −1.84385474550235854102929327946, −1.47416374860545609551734171743, −1.27636711695212601717034269464, −1.01051769371157704910862935187, 0, 0, 0, 0, 1.01051769371157704910862935187, 1.27636711695212601717034269464, 1.47416374860545609551734171743, 1.84385474550235854102929327946, 1.93204854835355817059576715164, 2.22285611311248012039573578777, 2.32672690624002495080543122039, 2.36184349748366255537354965017, 3.21180789664283586279885843703, 3.44467801549672841034346021419, 3.48002428696893587358138312408, 3.69208106538011908116223916334, 4.03708258876548845987557570353, 4.15070629559026359035880765486, 4.50115493962109934537005616677, 4.67341468752677984372186100092, 5.16677825462076977193601717194, 5.27786558896493820709254378740, 5.37413085475493896682548107564, 5.46791064519424492749877204619, 5.89245882144950373108338671439, 5.91277781717168167058725334013, 6.19773151345474869146988092202, 6.21849682748122146879098170753, 6.46837278826693760145734311562

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