Properties

Degree 18
Conductor $ 2^{27} \cdot 7^{9} \cdot 11^{9} \cdot 13^{9} $
Sign $-1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 9

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 4·5-s − 9·7-s − 11·9-s + 9·11-s + 9·13-s − 4·15-s − 11·17-s + 10·19-s − 9·21-s − 14·23-s − 15·25-s − 14·27-s − 10·29-s + 5·31-s + 9·33-s + 36·35-s − 16·37-s + 9·39-s + 2·41-s + 4·43-s + 44·45-s + 45·49-s − 11·51-s − 23·53-s − 36·55-s + 10·57-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.78·5-s − 3.40·7-s − 3.66·9-s + 2.71·11-s + 2.49·13-s − 1.03·15-s − 2.66·17-s + 2.29·19-s − 1.96·21-s − 2.91·23-s − 3·25-s − 2.69·27-s − 1.85·29-s + 0.898·31-s + 1.56·33-s + 6.08·35-s − 2.63·37-s + 1.44·39-s + 0.312·41-s + 0.609·43-s + 6.55·45-s + 45/7·49-s − 1.54·51-s − 3.15·53-s − 4.85·55-s + 1.32·57-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{27} \cdot 7^{9} \cdot 11^{9} \cdot 13^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr =\mathstrut & -\,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{27} \cdot 7^{9} \cdot 11^{9} \cdot 13^{9}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{9} \, L(s)\cr =\mathstrut & -\,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(18\)
\( N \)  =  \(2^{27} \cdot 7^{9} \cdot 11^{9} \cdot 13^{9}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{8008} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  9
Selberg data  =  $(18,\ 2^{27} \cdot 7^{9} \cdot 11^{9} \cdot 13^{9} ,\ ( \ : [1/2]^{9} ),\ -1 )$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;11,\;13\}$, \(F_p\) is a polynomial of degree 18. If $p \in \{2,\;7,\;11,\;13\}$, then $F_p$ is a polynomial of degree at most 17.
$p$$F_p$
bad2 \( 1 \)
7 \( ( 1 + T )^{9} \)
11 \( ( 1 - T )^{9} \)
13 \( ( 1 - T )^{9} \)
good3 \( 1 - T + 4 p T^{2} - p^{2} T^{3} + 25 p T^{4} - 41 T^{5} + 346 T^{6} - 161 T^{7} + 1289 T^{8} - 554 T^{9} + 1289 p T^{10} - 161 p^{2} T^{11} + 346 p^{3} T^{12} - 41 p^{4} T^{13} + 25 p^{6} T^{14} - p^{8} T^{15} + 4 p^{8} T^{16} - p^{8} T^{17} + p^{9} T^{18} \)
5 \( 1 + 4 T + 31 T^{2} + 22 p T^{3} + 483 T^{4} + 1473 T^{5} + 4837 T^{6} + 12551 T^{7} + 33717 T^{8} + 74369 T^{9} + 33717 p T^{10} + 12551 p^{2} T^{11} + 4837 p^{3} T^{12} + 1473 p^{4} T^{13} + 483 p^{5} T^{14} + 22 p^{7} T^{15} + 31 p^{7} T^{16} + 4 p^{8} T^{17} + p^{9} T^{18} \)
17 \( 1 + 11 T + 128 T^{2} + 877 T^{3} + 6361 T^{4} + 33933 T^{5} + 193412 T^{6} + 867961 T^{7} + 4213319 T^{8} + 16606726 T^{9} + 4213319 p T^{10} + 867961 p^{2} T^{11} + 193412 p^{3} T^{12} + 33933 p^{4} T^{13} + 6361 p^{5} T^{14} + 877 p^{6} T^{15} + 128 p^{7} T^{16} + 11 p^{8} T^{17} + p^{9} T^{18} \)
19 \( 1 - 10 T + 143 T^{2} - 1144 T^{3} + 9691 T^{4} - 62585 T^{5} + 400065 T^{6} - 2121153 T^{7} + 11016921 T^{8} - 48502067 T^{9} + 11016921 p T^{10} - 2121153 p^{2} T^{11} + 400065 p^{3} T^{12} - 62585 p^{4} T^{13} + 9691 p^{5} T^{14} - 1144 p^{6} T^{15} + 143 p^{7} T^{16} - 10 p^{8} T^{17} + p^{9} T^{18} \)
23 \( 1 + 14 T + 246 T^{2} + 2286 T^{3} + 1016 p T^{4} + 7161 p T^{5} + 1237044 T^{6} + 7013716 T^{7} + 41988877 T^{8} + 196159902 T^{9} + 41988877 p T^{10} + 7013716 p^{2} T^{11} + 1237044 p^{3} T^{12} + 7161 p^{5} T^{13} + 1016 p^{6} T^{14} + 2286 p^{6} T^{15} + 246 p^{7} T^{16} + 14 p^{8} T^{17} + p^{9} T^{18} \)
29 \( 1 + 10 T + 210 T^{2} + 1588 T^{3} + 19184 T^{4} + 116385 T^{5} + 1055760 T^{6} + 5377410 T^{7} + 40660657 T^{8} + 179445442 T^{9} + 40660657 p T^{10} + 5377410 p^{2} T^{11} + 1055760 p^{3} T^{12} + 116385 p^{4} T^{13} + 19184 p^{5} T^{14} + 1588 p^{6} T^{15} + 210 p^{7} T^{16} + 10 p^{8} T^{17} + p^{9} T^{18} \)
31 \( 1 - 5 T + 105 T^{2} - 658 T^{3} + 7331 T^{4} - 47729 T^{5} + 378881 T^{6} - 2212118 T^{7} + 15351802 T^{8} - 78722536 T^{9} + 15351802 p T^{10} - 2212118 p^{2} T^{11} + 378881 p^{3} T^{12} - 47729 p^{4} T^{13} + 7331 p^{5} T^{14} - 658 p^{6} T^{15} + 105 p^{7} T^{16} - 5 p^{8} T^{17} + p^{9} T^{18} \)
37 \( 1 + 16 T + 265 T^{2} + 2821 T^{3} + 30825 T^{4} + 265958 T^{5} + 2261395 T^{6} + 16306323 T^{7} + 115417966 T^{8} + 711331820 T^{9} + 115417966 p T^{10} + 16306323 p^{2} T^{11} + 2261395 p^{3} T^{12} + 265958 p^{4} T^{13} + 30825 p^{5} T^{14} + 2821 p^{6} T^{15} + 265 p^{7} T^{16} + 16 p^{8} T^{17} + p^{9} T^{18} \)
41 \( 1 - 2 T + 169 T^{2} - 579 T^{3} + 15219 T^{4} - 66790 T^{5} + 963781 T^{6} - 4783605 T^{7} + 1159186 p T^{8} - 235382632 T^{9} + 1159186 p^{2} T^{10} - 4783605 p^{2} T^{11} + 963781 p^{3} T^{12} - 66790 p^{4} T^{13} + 15219 p^{5} T^{14} - 579 p^{6} T^{15} + 169 p^{7} T^{16} - 2 p^{8} T^{17} + p^{9} T^{18} \)
43 \( 1 - 4 T + 294 T^{2} - 1178 T^{3} + 41242 T^{4} - 156793 T^{5} + 3623594 T^{6} - 12457441 T^{7} + 219052649 T^{8} - 651472055 T^{9} + 219052649 p T^{10} - 12457441 p^{2} T^{11} + 3623594 p^{3} T^{12} - 156793 p^{4} T^{13} + 41242 p^{5} T^{14} - 1178 p^{6} T^{15} + 294 p^{7} T^{16} - 4 p^{8} T^{17} + p^{9} T^{18} \)
47 \( 1 + 222 T^{2} - 498 T^{3} + 22064 T^{4} - 114863 T^{5} + 1342868 T^{6} - 11892176 T^{7} + 63370253 T^{8} - 713495866 T^{9} + 63370253 p T^{10} - 11892176 p^{2} T^{11} + 1342868 p^{3} T^{12} - 114863 p^{4} T^{13} + 22064 p^{5} T^{14} - 498 p^{6} T^{15} + 222 p^{7} T^{16} + p^{9} T^{18} \)
53 \( 1 + 23 T + 373 T^{2} + 3836 T^{3} + 30233 T^{4} + 153061 T^{5} + 385401 T^{6} - 2970078 T^{7} - 44656139 T^{8} - 401911747 T^{9} - 44656139 p T^{10} - 2970078 p^{2} T^{11} + 385401 p^{3} T^{12} + 153061 p^{4} T^{13} + 30233 p^{5} T^{14} + 3836 p^{6} T^{15} + 373 p^{7} T^{16} + 23 p^{8} T^{17} + p^{9} T^{18} \)
59 \( 1 - 9 T + 230 T^{2} - 1145 T^{3} + 18312 T^{4} - 37652 T^{5} + 1061744 T^{6} - 2277979 T^{7} + 88624297 T^{8} - 259246958 T^{9} + 88624297 p T^{10} - 2277979 p^{2} T^{11} + 1061744 p^{3} T^{12} - 37652 p^{4} T^{13} + 18312 p^{5} T^{14} - 1145 p^{6} T^{15} + 230 p^{7} T^{16} - 9 p^{8} T^{17} + p^{9} T^{18} \)
61 \( 1 + 14 T + 378 T^{2} + 3195 T^{3} + 52101 T^{4} + 260475 T^{5} + 3732090 T^{6} + 7203456 T^{7} + 183524293 T^{8} - 2793938 T^{9} + 183524293 p T^{10} + 7203456 p^{2} T^{11} + 3732090 p^{3} T^{12} + 260475 p^{4} T^{13} + 52101 p^{5} T^{14} + 3195 p^{6} T^{15} + 378 p^{7} T^{16} + 14 p^{8} T^{17} + p^{9} T^{18} \)
67 \( 1 - 8 T + 384 T^{2} - 2940 T^{3} + 74527 T^{4} - 530450 T^{5} + 141348 p T^{6} - 61039356 T^{7} + 859218535 T^{8} - 4862583934 T^{9} + 859218535 p T^{10} - 61039356 p^{2} T^{11} + 141348 p^{4} T^{12} - 530450 p^{4} T^{13} + 74527 p^{5} T^{14} - 2940 p^{6} T^{15} + 384 p^{7} T^{16} - 8 p^{8} T^{17} + p^{9} T^{18} \)
71 \( 1 + 20 T + 454 T^{2} + 7144 T^{3} + 102488 T^{4} + 1271930 T^{5} + 14515646 T^{6} + 149060480 T^{7} + 1421630731 T^{8} + 12439035124 T^{9} + 1421630731 p T^{10} + 149060480 p^{2} T^{11} + 14515646 p^{3} T^{12} + 1271930 p^{4} T^{13} + 102488 p^{5} T^{14} + 7144 p^{6} T^{15} + 454 p^{7} T^{16} + 20 p^{8} T^{17} + p^{9} T^{18} \)
73 \( 1 + 23 T + 525 T^{2} + 7059 T^{3} + 89094 T^{4} + 760469 T^{5} + 5999086 T^{6} + 26398007 T^{7} + 112110246 T^{8} - 65659504 T^{9} + 112110246 p T^{10} + 26398007 p^{2} T^{11} + 5999086 p^{3} T^{12} + 760469 p^{4} T^{13} + 89094 p^{5} T^{14} + 7059 p^{6} T^{15} + 525 p^{7} T^{16} + 23 p^{8} T^{17} + p^{9} T^{18} \)
79 \( 1 - 2 T + 374 T^{2} - 658 T^{3} + 78602 T^{4} - 106303 T^{5} + 11131368 T^{6} - 13006721 T^{7} + 1164179101 T^{8} - 1141949343 T^{9} + 1164179101 p T^{10} - 13006721 p^{2} T^{11} + 11131368 p^{3} T^{12} - 106303 p^{4} T^{13} + 78602 p^{5} T^{14} - 658 p^{6} T^{15} + 374 p^{7} T^{16} - 2 p^{8} T^{17} + p^{9} T^{18} \)
83 \( 1 + 9 T + 610 T^{2} + 4938 T^{3} + 174722 T^{4} + 1258235 T^{5} + 30717460 T^{6} + 193361618 T^{7} + 3645460843 T^{8} + 19557099679 T^{9} + 3645460843 p T^{10} + 193361618 p^{2} T^{11} + 30717460 p^{3} T^{12} + 1258235 p^{4} T^{13} + 174722 p^{5} T^{14} + 4938 p^{6} T^{15} + 610 p^{7} T^{16} + 9 p^{8} T^{17} + p^{9} T^{18} \)
89 \( 1 + 6 T + 677 T^{2} + 46 p T^{3} + 212857 T^{4} + 1228167 T^{5} + 40743411 T^{6} + 213245353 T^{7} + 5224210325 T^{8} + 23520533541 T^{9} + 5224210325 p T^{10} + 213245353 p^{2} T^{11} + 40743411 p^{3} T^{12} + 1228167 p^{4} T^{13} + 212857 p^{5} T^{14} + 46 p^{7} T^{15} + 677 p^{7} T^{16} + 6 p^{8} T^{17} + p^{9} T^{18} \)
97 \( 1 + 3 T + 491 T^{2} + 525 T^{3} + 120154 T^{4} - 3579 T^{5} + 19862938 T^{6} - 11754741 T^{7} + 2459126496 T^{8} - 1757418096 T^{9} + 2459126496 p T^{10} - 11754741 p^{2} T^{11} + 19862938 p^{3} T^{12} - 3579 p^{4} T^{13} + 120154 p^{5} T^{14} + 525 p^{6} T^{15} + 491 p^{7} T^{16} + 3 p^{8} T^{17} + p^{9} T^{18} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{18} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−3.23470863993461916772298950456, −3.18375683905111387990470460836, −3.17357111146747014823982874503, −3.13713166117487649157188235924, −3.13710020209251741921458513622, −2.95035979390530377935584761874, −2.94321831668259577596440716076, −2.56268395819645378475836752948, −2.50515502191842432925008762267, −2.43553338247583224016173300875, −2.27635525836250897698828010107, −2.24681862179079084309703973190, −2.16063502193691594456283304811, −2.15350763867920484572756021275, −2.12618314041601491093148987829, −2.01025317676713245820816027077, −1.69431119173604730834224934210, −1.53188313130641049401558120680, −1.33958762339205020865452462965, −1.32947067015154340542839994478, −1.24853594261325381188601390313, −1.22335425551692589712406716149, −1.04709616696701113969506369400, −0.971064677565740629932276614424, −0.872745076257913355056231590505, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.872745076257913355056231590505, 0.971064677565740629932276614424, 1.04709616696701113969506369400, 1.22335425551692589712406716149, 1.24853594261325381188601390313, 1.32947067015154340542839994478, 1.33958762339205020865452462965, 1.53188313130641049401558120680, 1.69431119173604730834224934210, 2.01025317676713245820816027077, 2.12618314041601491093148987829, 2.15350763867920484572756021275, 2.16063502193691594456283304811, 2.24681862179079084309703973190, 2.27635525836250897698828010107, 2.43553338247583224016173300875, 2.50515502191842432925008762267, 2.56268395819645378475836752948, 2.94321831668259577596440716076, 2.95035979390530377935584761874, 3.13710020209251741921458513622, 3.13713166117487649157188235924, 3.17357111146747014823982874503, 3.18375683905111387990470460836, 3.23470863993461916772298950456

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

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