Properties

Degree 18
Conductor $ 2^{18} \cdot 3^{9} \cdot 23^{9} \cdot 29^{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  − 9·3-s − 3·5-s + 7·7-s + 45·9-s − 2·11-s − 7·13-s + 27·15-s − 19-s − 63·21-s − 9·23-s − 17·25-s − 165·27-s + 9·29-s + 8·31-s + 18·33-s − 21·35-s − 8·37-s + 63·39-s − 19·41-s − 3·43-s − 135·45-s − 3·47-s − 16·49-s − 17·53-s + 6·55-s + 9·57-s − 10·59-s + ⋯
L(s)  = 1  − 5.19·3-s − 1.34·5-s + 2.64·7-s + 15·9-s − 0.603·11-s − 1.94·13-s + 6.97·15-s − 0.229·19-s − 13.7·21-s − 1.87·23-s − 3.39·25-s − 31.7·27-s + 1.67·29-s + 1.43·31-s + 3.13·33-s − 3.54·35-s − 1.31·37-s + 10.0·39-s − 2.96·41-s − 0.457·43-s − 20.1·45-s − 0.437·47-s − 2.28·49-s − 2.33·53-s + 0.809·55-s + 1.19·57-s − 1.30·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{9} \cdot 23^{9} \cdot 29^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{9} \cdot 23^{9} \cdot 29^{9}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(18\)
\( N \)  =  \(2^{18} \cdot 3^{9} \cdot 23^{9} \cdot 29^{9}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{8004} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(9\)
Selberg data  =  \((18,\ 2^{18} \cdot 3^{9} \cdot 23^{9} \cdot 29^{9} ,\ ( \ : [1/2]^{9} ),\ -1 )\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;23,\;29\}$,\(F_p(T)\) is a polynomial of degree 18. If $p \in \{2,\;3,\;23,\;29\}$, then $F_p(T)$ is a polynomial of degree at most 17.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( ( 1 + T )^{9} \)
23 \( ( 1 + T )^{9} \)
29 \( ( 1 - T )^{9} \)
good5 \( 1 + 3 T + 26 T^{2} + 76 T^{3} + 352 T^{4} + 897 T^{5} + 3136 T^{6} + 6906 T^{7} + 20281 T^{8} + 39416 T^{9} + 20281 p T^{10} + 6906 p^{2} T^{11} + 3136 p^{3} T^{12} + 897 p^{4} T^{13} + 352 p^{5} T^{14} + 76 p^{6} T^{15} + 26 p^{7} T^{16} + 3 p^{8} T^{17} + p^{9} T^{18} \)
7 \( 1 - p T + 65 T^{2} - 327 T^{3} + 1809 T^{4} - 1017 p T^{5} + 29153 T^{6} - 93123 T^{7} + 303428 T^{8} - 796948 T^{9} + 303428 p T^{10} - 93123 p^{2} T^{11} + 29153 p^{3} T^{12} - 1017 p^{5} T^{13} + 1809 p^{5} T^{14} - 327 p^{6} T^{15} + 65 p^{7} T^{16} - p^{9} T^{17} + p^{9} T^{18} \)
11 \( 1 + 2 T + 49 T^{2} + 95 T^{3} + 1244 T^{4} + 2309 T^{5} + 22092 T^{6} + 37857 T^{7} + 303948 T^{8} + 467906 T^{9} + 303948 p T^{10} + 37857 p^{2} T^{11} + 22092 p^{3} T^{12} + 2309 p^{4} T^{13} + 1244 p^{5} T^{14} + 95 p^{6} T^{15} + 49 p^{7} T^{16} + 2 p^{8} T^{17} + p^{9} T^{18} \)
13 \( 1 + 7 T + 87 T^{2} + 469 T^{3} + 3480 T^{4} + 15664 T^{5} + 87354 T^{6} + 337444 T^{7} + 9131 p^{2} T^{8} + 5151761 T^{9} + 9131 p^{3} T^{10} + 337444 p^{2} T^{11} + 87354 p^{3} T^{12} + 15664 p^{4} T^{13} + 3480 p^{5} T^{14} + 469 p^{6} T^{15} + 87 p^{7} T^{16} + 7 p^{8} T^{17} + p^{9} T^{18} \)
17 \( 1 + 98 T^{2} - 115 T^{3} + 4525 T^{4} - 8601 T^{5} + 140632 T^{6} - 281898 T^{7} + 3253471 T^{8} - 5743037 T^{9} + 3253471 p T^{10} - 281898 p^{2} T^{11} + 140632 p^{3} T^{12} - 8601 p^{4} T^{13} + 4525 p^{5} T^{14} - 115 p^{6} T^{15} + 98 p^{7} T^{16} + p^{9} T^{18} \)
19 \( 1 + T + 97 T^{2} + 6 p T^{3} + 4774 T^{4} + 6136 T^{5} + 160280 T^{6} + 10452 p T^{7} + 4006913 T^{8} + 4421409 T^{9} + 4006913 p T^{10} + 10452 p^{3} T^{11} + 160280 p^{3} T^{12} + 6136 p^{4} T^{13} + 4774 p^{5} T^{14} + 6 p^{7} T^{15} + 97 p^{7} T^{16} + p^{8} T^{17} + p^{9} T^{18} \)
31 \( 1 - 8 T + 153 T^{2} - 1097 T^{3} + 12326 T^{4} - 78609 T^{5} + 662686 T^{6} - 3844949 T^{7} + 26675000 T^{8} - 137552206 T^{9} + 26675000 p T^{10} - 3844949 p^{2} T^{11} + 662686 p^{3} T^{12} - 78609 p^{4} T^{13} + 12326 p^{5} T^{14} - 1097 p^{6} T^{15} + 153 p^{7} T^{16} - 8 p^{8} T^{17} + p^{9} T^{18} \)
37 \( 1 + 8 T + 253 T^{2} + 1765 T^{3} + 30221 T^{4} + 186187 T^{5} + 2251317 T^{6} + 329446 p T^{7} + 115890110 T^{8} + 541298595 T^{9} + 115890110 p T^{10} + 329446 p^{3} T^{11} + 2251317 p^{3} T^{12} + 186187 p^{4} T^{13} + 30221 p^{5} T^{14} + 1765 p^{6} T^{15} + 253 p^{7} T^{16} + 8 p^{8} T^{17} + p^{9} T^{18} \)
41 \( 1 + 19 T + 400 T^{2} + 4888 T^{3} + 60100 T^{4} + 550873 T^{5} + 5032972 T^{6} + 37539398 T^{7} + 281599251 T^{8} + 1792540472 T^{9} + 281599251 p T^{10} + 37539398 p^{2} T^{11} + 5032972 p^{3} T^{12} + 550873 p^{4} T^{13} + 60100 p^{5} T^{14} + 4888 p^{6} T^{15} + 400 p^{7} T^{16} + 19 p^{8} T^{17} + p^{9} T^{18} \)
43 \( 1 + 3 T + 120 T^{2} + 800 T^{3} + 10631 T^{4} + 70060 T^{5} + 768666 T^{6} + 4485320 T^{7} + 39922183 T^{8} + 230952547 T^{9} + 39922183 p T^{10} + 4485320 p^{2} T^{11} + 768666 p^{3} T^{12} + 70060 p^{4} T^{13} + 10631 p^{5} T^{14} + 800 p^{6} T^{15} + 120 p^{7} T^{16} + 3 p^{8} T^{17} + p^{9} T^{18} \)
47 \( 1 + 3 T + 278 T^{2} + 977 T^{3} + 36995 T^{4} + 144455 T^{5} + 3167707 T^{6} + 12728693 T^{7} + 196210413 T^{8} + 731862316 T^{9} + 196210413 p T^{10} + 12728693 p^{2} T^{11} + 3167707 p^{3} T^{12} + 144455 p^{4} T^{13} + 36995 p^{5} T^{14} + 977 p^{6} T^{15} + 278 p^{7} T^{16} + 3 p^{8} T^{17} + p^{9} T^{18} \)
53 \( 1 + 17 T + 398 T^{2} + 4052 T^{3} + 52736 T^{4} + 348193 T^{5} + 3384098 T^{6} + 14443272 T^{7} + 145879247 T^{8} + 516370164 T^{9} + 145879247 p T^{10} + 14443272 p^{2} T^{11} + 3384098 p^{3} T^{12} + 348193 p^{4} T^{13} + 52736 p^{5} T^{14} + 4052 p^{6} T^{15} + 398 p^{7} T^{16} + 17 p^{8} T^{17} + p^{9} T^{18} \)
59 \( 1 + 10 T + 427 T^{2} + 3704 T^{3} + 86119 T^{4} + 648418 T^{5} + 10694205 T^{6} + 69416192 T^{7} + 899209120 T^{8} + 4957696952 T^{9} + 899209120 p T^{10} + 69416192 p^{2} T^{11} + 10694205 p^{3} T^{12} + 648418 p^{4} T^{13} + 86119 p^{5} T^{14} + 3704 p^{6} T^{15} + 427 p^{7} T^{16} + 10 p^{8} T^{17} + p^{9} T^{18} \)
61 \( 1 - T + 415 T^{2} - 281 T^{3} + 79991 T^{4} - 31893 T^{5} + 9569195 T^{6} - 1981755 T^{7} + 797704854 T^{8} - 104297468 T^{9} + 797704854 p T^{10} - 1981755 p^{2} T^{11} + 9569195 p^{3} T^{12} - 31893 p^{4} T^{13} + 79991 p^{5} T^{14} - 281 p^{6} T^{15} + 415 p^{7} T^{16} - p^{8} T^{17} + p^{9} T^{18} \)
67 \( 1 - 12 T + 274 T^{2} - 3235 T^{3} + 47142 T^{4} - 446605 T^{5} + 5204776 T^{6} - 44226721 T^{7} + 436542763 T^{8} - 3279783782 T^{9} + 436542763 p T^{10} - 44226721 p^{2} T^{11} + 5204776 p^{3} T^{12} - 446605 p^{4} T^{13} + 47142 p^{5} T^{14} - 3235 p^{6} T^{15} + 274 p^{7} T^{16} - 12 p^{8} T^{17} + p^{9} T^{18} \)
71 \( 1 + 7 T + 533 T^{2} + 3329 T^{3} + 132072 T^{4} + 728958 T^{5} + 19985738 T^{6} + 95991266 T^{7} + 2034708327 T^{8} + 8308609245 T^{9} + 2034708327 p T^{10} + 95991266 p^{2} T^{11} + 19985738 p^{3} T^{12} + 728958 p^{4} T^{13} + 132072 p^{5} T^{14} + 3329 p^{6} T^{15} + 533 p^{7} T^{16} + 7 p^{8} T^{17} + p^{9} T^{18} \)
73 \( 1 - 13 T + 506 T^{2} - 4747 T^{3} + 107167 T^{4} - 734909 T^{5} + 13144645 T^{6} - 67606851 T^{7} + 1145412177 T^{8} - 4992931436 T^{9} + 1145412177 p T^{10} - 67606851 p^{2} T^{11} + 13144645 p^{3} T^{12} - 734909 p^{4} T^{13} + 107167 p^{5} T^{14} - 4747 p^{6} T^{15} + 506 p^{7} T^{16} - 13 p^{8} T^{17} + p^{9} T^{18} \)
79 \( 1 + 10 T + 425 T^{2} + 2715 T^{3} + 76957 T^{4} + 321871 T^{5} + 9247137 T^{6} + 29782950 T^{7} + 906879556 T^{8} + 2586333853 T^{9} + 906879556 p T^{10} + 29782950 p^{2} T^{11} + 9247137 p^{3} T^{12} + 321871 p^{4} T^{13} + 76957 p^{5} T^{14} + 2715 p^{6} T^{15} + 425 p^{7} T^{16} + 10 p^{8} T^{17} + p^{9} T^{18} \)
83 \( 1 - 9 T + 422 T^{2} - 4342 T^{3} + 86772 T^{4} - 940635 T^{5} + 11854664 T^{6} - 126639668 T^{7} + 1224295309 T^{8} - 12173194872 T^{9} + 1224295309 p T^{10} - 126639668 p^{2} T^{11} + 11854664 p^{3} T^{12} - 940635 p^{4} T^{13} + 86772 p^{5} T^{14} - 4342 p^{6} T^{15} + 422 p^{7} T^{16} - 9 p^{8} T^{17} + p^{9} T^{18} \)
89 \( 1 + 5 T + 291 T^{2} + 2317 T^{3} + 51986 T^{4} + 466616 T^{5} + 7203344 T^{6} + 61936092 T^{7} + 797299301 T^{8} + 6209910283 T^{9} + 797299301 p T^{10} + 61936092 p^{2} T^{11} + 7203344 p^{3} T^{12} + 466616 p^{4} T^{13} + 51986 p^{5} T^{14} + 2317 p^{6} T^{15} + 291 p^{7} T^{16} + 5 p^{8} T^{17} + p^{9} T^{18} \)
97 \( 1 + 7 T + 269 T^{2} + 2553 T^{3} + 38361 T^{4} + 294733 T^{5} + 3831941 T^{6} + 15532083 T^{7} + 260082212 T^{8} + 872400240 T^{9} + 260082212 p T^{10} + 15532083 p^{2} T^{11} + 3831941 p^{3} T^{12} + 294733 p^{4} T^{13} + 38361 p^{5} T^{14} + 2553 p^{6} T^{15} + 269 p^{7} T^{16} + 7 p^{8} T^{17} + p^{9} T^{18} \)
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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

−3.25833081807587787369817275344, −3.22204000997571439855715330273, −3.21295244883324497717281458323, −3.12659056681963479877969488030, −3.05498929682246183837394596204, −2.89028506430268042531286233790, −2.85326570971119426892813440256, −2.38578916702211903478329458008, −2.23370913571935563586287951206, −2.22486212512082637675448942485, −2.20027547337030882009090717352, −2.10294238094845670970731899233, −2.08031465828343288149424535739, −2.06991288056023205278420756403, −2.04762520381333540327796156913, −1.89625518394172723780987431180, −1.49253465800154112722354071227, −1.31904606339320929522800819799, −1.29771476150099003287640726054, −1.27063393933418541834367086869, −1.24680424123389071675228460127, −1.18024230486770180626745678985, −1.17371600385289728730374795885, −0.992089007376262196524023425623, −0.828119212387202146088289562628, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.828119212387202146088289562628, 0.992089007376262196524023425623, 1.17371600385289728730374795885, 1.18024230486770180626745678985, 1.24680424123389071675228460127, 1.27063393933418541834367086869, 1.29771476150099003287640726054, 1.31904606339320929522800819799, 1.49253465800154112722354071227, 1.89625518394172723780987431180, 2.04762520381333540327796156913, 2.06991288056023205278420756403, 2.08031465828343288149424535739, 2.10294238094845670970731899233, 2.20027547337030882009090717352, 2.22486212512082637675448942485, 2.23370913571935563586287951206, 2.38578916702211903478329458008, 2.85326570971119426892813440256, 2.89028506430268042531286233790, 3.05498929682246183837394596204, 3.12659056681963479877969488030, 3.21295244883324497717281458323, 3.22204000997571439855715330273, 3.25833081807587787369817275344

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

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