Properties

Degree 18
Conductor $ 2^{27} \cdot 3^{9} \cdot 167^{9} $
Sign $-1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 9

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 9·3-s − 6·5-s − 11·7-s + 45·9-s + 11-s − 4·13-s − 54·15-s − 9·17-s − 8·19-s − 99·21-s − 7·23-s − 5·25-s + 165·27-s − 9·29-s − 25·31-s + 9·33-s + 66·35-s − 6·37-s − 36·39-s − 4·41-s − 24·43-s − 270·45-s − 16·47-s + 31·49-s − 81·51-s − 26·53-s − 6·55-s + ⋯
L(s)  = 1  + 5.19·3-s − 2.68·5-s − 4.15·7-s + 15·9-s + 0.301·11-s − 1.10·13-s − 13.9·15-s − 2.18·17-s − 1.83·19-s − 21.6·21-s − 1.45·23-s − 25-s + 31.7·27-s − 1.67·29-s − 4.49·31-s + 1.56·33-s + 11.1·35-s − 0.986·37-s − 5.76·39-s − 0.624·41-s − 3.65·43-s − 40.2·45-s − 2.33·47-s + 31/7·49-s − 11.3·51-s − 3.57·53-s − 0.809·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{27} \cdot 3^{9} \cdot 167^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr =\mathstrut & -\,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{27} \cdot 3^{9} \cdot 167^{9}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{9} \, L(s)\cr =\mathstrut & -\,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(18\)
\( N \)  =  \(2^{27} \cdot 3^{9} \cdot 167^{9}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{4008} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  9
Selberg data  =  $(18,\ 2^{27} \cdot 3^{9} \cdot 167^{9} ,\ ( \ : [1/2]^{9} ),\ -1 )$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;167\}$, \(F_p\) is a polynomial of degree 18. If $p \in \{2,\;3,\;167\}$, then $F_p$ is a polynomial of degree at most 17.
$p$$F_p$
bad2 \( 1 \)
3 \( ( 1 - T )^{9} \)
167 \( ( 1 + T )^{9} \)
good5 \( 1 + 6 T + 41 T^{2} + 173 T^{3} + 719 T^{4} + 478 p T^{5} + 7554 T^{6} + 20743 T^{7} + 10721 p T^{8} + 123718 T^{9} + 10721 p^{2} T^{10} + 20743 p^{2} T^{11} + 7554 p^{3} T^{12} + 478 p^{5} T^{13} + 719 p^{5} T^{14} + 173 p^{6} T^{15} + 41 p^{7} T^{16} + 6 p^{8} T^{17} + p^{9} T^{18} \)
7 \( 1 + 11 T + 90 T^{2} + 528 T^{3} + 2641 T^{4} + 11154 T^{5} + 42250 T^{6} + 141980 T^{7} + 62250 p T^{8} + 1204094 T^{9} + 62250 p^{2} T^{10} + 141980 p^{2} T^{11} + 42250 p^{3} T^{12} + 11154 p^{4} T^{13} + 2641 p^{5} T^{14} + 528 p^{6} T^{15} + 90 p^{7} T^{16} + 11 p^{8} T^{17} + p^{9} T^{18} \)
11 \( 1 - T + 52 T^{2} - 70 T^{3} + 1332 T^{4} - 2129 T^{5} + 23490 T^{6} - 38142 T^{7} + 322805 T^{8} - 481364 T^{9} + 322805 p T^{10} - 38142 p^{2} T^{11} + 23490 p^{3} T^{12} - 2129 p^{4} T^{13} + 1332 p^{5} T^{14} - 70 p^{6} T^{15} + 52 p^{7} T^{16} - p^{8} T^{17} + p^{9} T^{18} \)
13 \( 1 + 4 T + 74 T^{2} + 264 T^{3} + 2783 T^{4} + 8611 T^{5} + 67581 T^{6} + 182788 T^{7} + 1176589 T^{8} + 2768138 T^{9} + 1176589 p T^{10} + 182788 p^{2} T^{11} + 67581 p^{3} T^{12} + 8611 p^{4} T^{13} + 2783 p^{5} T^{14} + 264 p^{6} T^{15} + 74 p^{7} T^{16} + 4 p^{8} T^{17} + p^{9} T^{18} \)
17 \( 1 + 9 T + 105 T^{2} + 636 T^{3} + 4431 T^{4} + 20141 T^{5} + 107753 T^{6} + 398006 T^{7} + 1917132 T^{8} + 6639284 T^{9} + 1917132 p T^{10} + 398006 p^{2} T^{11} + 107753 p^{3} T^{12} + 20141 p^{4} T^{13} + 4431 p^{5} T^{14} + 636 p^{6} T^{15} + 105 p^{7} T^{16} + 9 p^{8} T^{17} + p^{9} T^{18} \)
19 \( 1 + 8 T + 74 T^{2} + 18 p T^{3} + 1683 T^{4} + 3555 T^{5} + 13241 T^{6} - 21770 T^{7} + 26115 T^{8} - 779550 T^{9} + 26115 p T^{10} - 21770 p^{2} T^{11} + 13241 p^{3} T^{12} + 3555 p^{4} T^{13} + 1683 p^{5} T^{14} + 18 p^{7} T^{15} + 74 p^{7} T^{16} + 8 p^{8} T^{17} + p^{9} T^{18} \)
23 \( 1 + 7 T + 122 T^{2} + 616 T^{3} + 6996 T^{4} + 30587 T^{5} + 281820 T^{6} + 1091516 T^{7} + 8438553 T^{8} + 28681556 T^{9} + 8438553 p T^{10} + 1091516 p^{2} T^{11} + 281820 p^{3} T^{12} + 30587 p^{4} T^{13} + 6996 p^{5} T^{14} + 616 p^{6} T^{15} + 122 p^{7} T^{16} + 7 p^{8} T^{17} + p^{9} T^{18} \)
29 \( 1 + 9 T + 160 T^{2} + 954 T^{3} + 10872 T^{4} + 50409 T^{5} + 484084 T^{6} + 1856662 T^{7} + 16472499 T^{8} + 56162028 T^{9} + 16472499 p T^{10} + 1856662 p^{2} T^{11} + 484084 p^{3} T^{12} + 50409 p^{4} T^{13} + 10872 p^{5} T^{14} + 954 p^{6} T^{15} + 160 p^{7} T^{16} + 9 p^{8} T^{17} + p^{9} T^{18} \)
31 \( 1 + 25 T + 431 T^{2} + 5443 T^{3} + 57907 T^{4} + 525407 T^{5} + 4233640 T^{6} + 30306973 T^{7} + 196210743 T^{8} + 1145270768 T^{9} + 196210743 p T^{10} + 30306973 p^{2} T^{11} + 4233640 p^{3} T^{12} + 525407 p^{4} T^{13} + 57907 p^{5} T^{14} + 5443 p^{6} T^{15} + 431 p^{7} T^{16} + 25 p^{8} T^{17} + p^{9} T^{18} \)
37 \( 1 + 6 T + 165 T^{2} + 967 T^{3} + 12241 T^{4} + 69142 T^{5} + 530842 T^{6} + 3130351 T^{7} + 16937275 T^{8} + 117767388 T^{9} + 16937275 p T^{10} + 3130351 p^{2} T^{11} + 530842 p^{3} T^{12} + 69142 p^{4} T^{13} + 12241 p^{5} T^{14} + 967 p^{6} T^{15} + 165 p^{7} T^{16} + 6 p^{8} T^{17} + p^{9} T^{18} \)
41 \( 1 + 4 T + 257 T^{2} + 1351 T^{3} + 31614 T^{4} + 184863 T^{5} + 2521094 T^{6} + 14274085 T^{7} + 142940352 T^{8} + 712583638 T^{9} + 142940352 p T^{10} + 14274085 p^{2} T^{11} + 2521094 p^{3} T^{12} + 184863 p^{4} T^{13} + 31614 p^{5} T^{14} + 1351 p^{6} T^{15} + 257 p^{7} T^{16} + 4 p^{8} T^{17} + p^{9} T^{18} \)
43 \( 1 + 24 T + 515 T^{2} + 7423 T^{3} + 96980 T^{4} + 1029695 T^{5} + 10046614 T^{6} + 84278929 T^{7} + 654510506 T^{8} + 4455064042 T^{9} + 654510506 p T^{10} + 84278929 p^{2} T^{11} + 10046614 p^{3} T^{12} + 1029695 p^{4} T^{13} + 96980 p^{5} T^{14} + 7423 p^{6} T^{15} + 515 p^{7} T^{16} + 24 p^{8} T^{17} + p^{9} T^{18} \)
47 \( 1 + 16 T + 443 T^{2} + 5414 T^{3} + 84955 T^{4} + 827727 T^{5} + 9385058 T^{6} + 74534126 T^{7} + 662991021 T^{8} + 4317365814 T^{9} + 662991021 p T^{10} + 74534126 p^{2} T^{11} + 9385058 p^{3} T^{12} + 827727 p^{4} T^{13} + 84955 p^{5} T^{14} + 5414 p^{6} T^{15} + 443 p^{7} T^{16} + 16 p^{8} T^{17} + p^{9} T^{18} \)
53 \( 1 + 26 T + 527 T^{2} + 7026 T^{3} + 77435 T^{4} + 638817 T^{5} + 4321678 T^{6} + 20643772 T^{7} + 78929197 T^{8} + 281350616 T^{9} + 78929197 p T^{10} + 20643772 p^{2} T^{11} + 4321678 p^{3} T^{12} + 638817 p^{4} T^{13} + 77435 p^{5} T^{14} + 7026 p^{6} T^{15} + 527 p^{7} T^{16} + 26 p^{8} T^{17} + p^{9} T^{18} \)
59 \( 1 + 4 T + 325 T^{2} + 1542 T^{3} + 55375 T^{4} + 257845 T^{5} + 6265316 T^{6} + 26719662 T^{7} + 503343089 T^{8} + 1894242454 T^{9} + 503343089 p T^{10} + 26719662 p^{2} T^{11} + 6265316 p^{3} T^{12} + 257845 p^{4} T^{13} + 55375 p^{5} T^{14} + 1542 p^{6} T^{15} + 325 p^{7} T^{16} + 4 p^{8} T^{17} + p^{9} T^{18} \)
61 \( 1 + 20 T + 602 T^{2} + 9248 T^{3} + 155243 T^{4} + 1900323 T^{5} + 22802367 T^{6} + 226604280 T^{7} + 2116420563 T^{8} + 17147925490 T^{9} + 2116420563 p T^{10} + 226604280 p^{2} T^{11} + 22802367 p^{3} T^{12} + 1900323 p^{4} T^{13} + 155243 p^{5} T^{14} + 9248 p^{6} T^{15} + 602 p^{7} T^{16} + 20 p^{8} T^{17} + p^{9} T^{18} \)
67 \( 1 + 25 T + 568 T^{2} + 9209 T^{3} + 132959 T^{4} + 1619633 T^{5} + 18110518 T^{6} + 180254541 T^{7} + 1671954472 T^{8} + 14144938314 T^{9} + 1671954472 p T^{10} + 180254541 p^{2} T^{11} + 18110518 p^{3} T^{12} + 1619633 p^{4} T^{13} + 132959 p^{5} T^{14} + 9209 p^{6} T^{15} + 568 p^{7} T^{16} + 25 p^{8} T^{17} + p^{9} T^{18} \)
71 \( 1 + 15 T + 449 T^{2} + 5499 T^{3} + 91049 T^{4} + 954797 T^{5} + 11431765 T^{6} + 106092249 T^{7} + 1035484196 T^{8} + 8617953600 T^{9} + 1035484196 p T^{10} + 106092249 p^{2} T^{11} + 11431765 p^{3} T^{12} + 954797 p^{4} T^{13} + 91049 p^{5} T^{14} + 5499 p^{6} T^{15} + 449 p^{7} T^{16} + 15 p^{8} T^{17} + p^{9} T^{18} \)
73 \( 1 + 10 T + 276 T^{2} + 3380 T^{3} + 53683 T^{4} + 589381 T^{5} + 6867531 T^{6} + 71234924 T^{7} + 666366469 T^{8} + 5992553762 T^{9} + 666366469 p T^{10} + 71234924 p^{2} T^{11} + 6867531 p^{3} T^{12} + 589381 p^{4} T^{13} + 53683 p^{5} T^{14} + 3380 p^{6} T^{15} + 276 p^{7} T^{16} + 10 p^{8} T^{17} + p^{9} T^{18} \)
79 \( 1 + 34 T + 941 T^{2} + 18473 T^{3} + 313218 T^{4} + 4495933 T^{5} + 57476662 T^{6} + 655683215 T^{7} + 6748904942 T^{8} + 63046688450 T^{9} + 6748904942 p T^{10} + 655683215 p^{2} T^{11} + 57476662 p^{3} T^{12} + 4495933 p^{4} T^{13} + 313218 p^{5} T^{14} + 18473 p^{6} T^{15} + 941 p^{7} T^{16} + 34 p^{8} T^{17} + p^{9} T^{18} \)
83 \( 1 + 4 T + 451 T^{2} + 512 T^{3} + 86177 T^{4} - 2635 p T^{5} + 9317210 T^{6} - 68434988 T^{7} + 730048475 T^{8} - 8141053038 T^{9} + 730048475 p T^{10} - 68434988 p^{2} T^{11} + 9317210 p^{3} T^{12} - 2635 p^{5} T^{13} + 86177 p^{5} T^{14} + 512 p^{6} T^{15} + 451 p^{7} T^{16} + 4 p^{8} T^{17} + p^{9} T^{18} \)
89 \( 1 - 13 T + 480 T^{2} - 6469 T^{3} + 126713 T^{4} - 1525667 T^{5} + 22163950 T^{6} - 229481013 T^{7} + 2736499528 T^{8} - 24097791692 T^{9} + 2736499528 p T^{10} - 229481013 p^{2} T^{11} + 22163950 p^{3} T^{12} - 1525667 p^{4} T^{13} + 126713 p^{5} T^{14} - 6469 p^{6} T^{15} + 480 p^{7} T^{16} - 13 p^{8} T^{17} + p^{9} T^{18} \)
97 \( 1 + 4 T + 306 T^{2} + 941 T^{3} + 54811 T^{4} + 87444 T^{5} + 7554000 T^{6} + 3292913 T^{7} + 811591682 T^{8} - 7315132 T^{9} + 811591682 p T^{10} + 3292913 p^{2} T^{11} + 7554000 p^{3} T^{12} + 87444 p^{4} T^{13} + 54811 p^{5} T^{14} + 941 p^{6} T^{15} + 306 p^{7} T^{16} + 4 p^{8} T^{17} + p^{9} T^{18} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{18} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−3.54870421830755294067481489400, −3.52790058170199856889482901447, −3.42954198585822308054046392617, −3.36650408746486904357079300050, −3.22924559491694813465683048314, −3.19355338488292817630320468877, −3.15411019598286334172903177789, −3.11930735800683632948912094801, −2.74681037654593591062045816033, −2.61532274271596415639689815077, −2.60225418646416617478139066579, −2.58945563122126296250162429783, −2.53260671561345394896616841098, −2.35119402809370291308673280722, −2.33776850107007713865957999060, −2.24303317049076766486315318735, −1.81871984208376143198129717124, −1.75956024153102106506725367570, −1.68017941238334238094147569791, −1.63316053981271216236284161298, −1.59523734853418015635971086764, −1.51092403242314701006793699660, −1.46454003648064016646555818374, −1.30034461778575977840305754336, −1.25977988748857732524129212862, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.25977988748857732524129212862, 1.30034461778575977840305754336, 1.46454003648064016646555818374, 1.51092403242314701006793699660, 1.59523734853418015635971086764, 1.63316053981271216236284161298, 1.68017941238334238094147569791, 1.75956024153102106506725367570, 1.81871984208376143198129717124, 2.24303317049076766486315318735, 2.33776850107007713865957999060, 2.35119402809370291308673280722, 2.53260671561345394896616841098, 2.58945563122126296250162429783, 2.60225418646416617478139066579, 2.61532274271596415639689815077, 2.74681037654593591062045816033, 3.11930735800683632948912094801, 3.15411019598286334172903177789, 3.19355338488292817630320468877, 3.22924559491694813465683048314, 3.36650408746486904357079300050, 3.42954198585822308054046392617, 3.52790058170199856889482901447, 3.54870421830755294067481489400

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

Plot not available for L-functions of degree greater than 10.