Properties

Label 18-6018e9-1.1-c1e9-0-2
Degree $18$
Conductor $1.035\times 10^{34}$
Sign $-1$
Analytic cond. $1.36635\times 10^{15}$
Root an. cond. $6.93209$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $9$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 9·2-s − 9·3-s + 45·4-s + 6·5-s + 81·6-s − 11·7-s − 165·8-s + 45·9-s − 54·10-s + 11-s − 405·12-s + 4·13-s + 99·14-s − 54·15-s + 495·16-s − 9·17-s − 405·18-s − 13·19-s + 270·20-s + 99·21-s − 9·22-s − 6·23-s + 1.48e3·24-s − 36·26-s − 165·27-s − 495·28-s + 10·29-s + ⋯
L(s)  = 1  − 6.36·2-s − 5.19·3-s + 45/2·4-s + 2.68·5-s + 33.0·6-s − 4.15·7-s − 58.3·8-s + 15·9-s − 17.0·10-s + 0.301·11-s − 116.·12-s + 1.10·13-s + 26.4·14-s − 13.9·15-s + 123.·16-s − 2.18·17-s − 95.4·18-s − 2.98·19-s + 60.3·20-s + 21.6·21-s − 1.91·22-s − 1.25·23-s + 303.·24-s − 7.06·26-s − 31.7·27-s − 93.5·28-s + 1.85·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{9} \cdot 17^{9} \cdot 59^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{9} \cdot 17^{9} \cdot 59^{9}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(18\)
Conductor: \(2^{9} \cdot 3^{9} \cdot 17^{9} \cdot 59^{9}\)
Sign: $-1$
Analytic conductor: \(1.36635\times 10^{15}\)
Root analytic conductor: \(6.93209\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(9\)
Selberg data: \((18,\ 2^{9} \cdot 3^{9} \cdot 17^{9} \cdot 59^{9} ,\ ( \ : [1/2]^{9} ),\ -1 )\)

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$$F_p(T)$
bad2 \( ( 1 + T )^{9} \)
3 \( ( 1 + T )^{9} \)
17 \( ( 1 + T )^{9} \)
59 \( ( 1 + T )^{9} \)
good5 \( 1 - 6 T + 36 T^{2} - 27 p T^{3} + 19 p^{2} T^{4} - 1339 T^{5} + 3607 T^{6} - 8598 T^{7} + 20449 T^{8} - 45204 T^{9} + 20449 p T^{10} - 8598 p^{2} T^{11} + 3607 p^{3} T^{12} - 1339 p^{4} T^{13} + 19 p^{7} T^{14} - 27 p^{7} T^{15} + 36 p^{7} T^{16} - 6 p^{8} T^{17} + p^{9} T^{18} \)
7 \( 1 + 11 T + 13 p T^{2} + 545 T^{3} + 2776 T^{4} + 11887 T^{5} + 45333 T^{6} + 152805 T^{7} + 468746 T^{8} + 1295394 T^{9} + 468746 p T^{10} + 152805 p^{2} T^{11} + 45333 p^{3} T^{12} + 11887 p^{4} T^{13} + 2776 p^{5} T^{14} + 545 p^{6} T^{15} + 13 p^{8} T^{16} + 11 p^{8} T^{17} + p^{9} T^{18} \)
11 \( 1 - T + 5 p T^{2} - 78 T^{3} + 1586 T^{4} - 2543 T^{5} + 2803 p T^{6} - 50189 T^{7} + 40225 p T^{8} - 665306 T^{9} + 40225 p^{2} T^{10} - 50189 p^{2} T^{11} + 2803 p^{4} T^{12} - 2543 p^{4} T^{13} + 1586 p^{5} T^{14} - 78 p^{6} T^{15} + 5 p^{8} T^{16} - p^{8} T^{17} + p^{9} T^{18} \)
13 \( 1 - 4 T + 54 T^{2} - 290 T^{3} + 1874 T^{4} - 8878 T^{5} + 46560 T^{6} - 186326 T^{7} + 796527 T^{8} - 2883916 T^{9} + 796527 p T^{10} - 186326 p^{2} T^{11} + 46560 p^{3} T^{12} - 8878 p^{4} T^{13} + 1874 p^{5} T^{14} - 290 p^{6} T^{15} + 54 p^{7} T^{16} - 4 p^{8} T^{17} + p^{9} T^{18} \)
19 \( 1 + 13 T + 159 T^{2} + 1170 T^{3} + 7957 T^{4} + 37685 T^{5} + 168393 T^{6} + 496718 T^{7} + 1691010 T^{8} + 227844 p T^{9} + 1691010 p T^{10} + 496718 p^{2} T^{11} + 168393 p^{3} T^{12} + 37685 p^{4} T^{13} + 7957 p^{5} T^{14} + 1170 p^{6} T^{15} + 159 p^{7} T^{16} + 13 p^{8} T^{17} + p^{9} T^{18} \)
23 \( 1 + 6 T + 132 T^{2} + 764 T^{3} + 8943 T^{4} + 46939 T^{5} + 397909 T^{6} + 1833964 T^{7} + 12541318 T^{8} + 49942456 T^{9} + 12541318 p T^{10} + 1833964 p^{2} T^{11} + 397909 p^{3} T^{12} + 46939 p^{4} T^{13} + 8943 p^{5} T^{14} + 764 p^{6} T^{15} + 132 p^{7} T^{16} + 6 p^{8} T^{17} + p^{9} T^{18} \)
29 \( 1 - 10 T + 154 T^{2} - 1535 T^{3} + 14289 T^{4} - 112663 T^{5} + 856855 T^{6} - 5526672 T^{7} + 34557483 T^{8} - 191504420 T^{9} + 34557483 p T^{10} - 5526672 p^{2} T^{11} + 856855 p^{3} T^{12} - 112663 p^{4} T^{13} + 14289 p^{5} T^{14} - 1535 p^{6} T^{15} + 154 p^{7} T^{16} - 10 p^{8} T^{17} + p^{9} T^{18} \)
31 \( 1 - T + 135 T^{2} + 36 T^{3} + 9101 T^{4} + 8582 T^{5} + 14257 p T^{6} + 429396 T^{7} + 17228920 T^{8} + 14081542 T^{9} + 17228920 p T^{10} + 429396 p^{2} T^{11} + 14257 p^{4} T^{12} + 8582 p^{4} T^{13} + 9101 p^{5} T^{14} + 36 p^{6} T^{15} + 135 p^{7} T^{16} - p^{8} T^{17} + p^{9} T^{18} \)
37 \( 1 + 2 T + 188 T^{2} + 423 T^{3} + 18227 T^{4} + 44165 T^{5} + 1209239 T^{6} + 2807696 T^{7} + 59254779 T^{8} + 122227240 T^{9} + 59254779 p T^{10} + 2807696 p^{2} T^{11} + 1209239 p^{3} T^{12} + 44165 p^{4} T^{13} + 18227 p^{5} T^{14} + 423 p^{6} T^{15} + 188 p^{7} T^{16} + 2 p^{8} T^{17} + p^{9} T^{18} \)
41 \( 1 - 20 T + 392 T^{2} - 5020 T^{3} + 61165 T^{4} - 593867 T^{5} + 5495533 T^{6} - 43187704 T^{7} + 324961130 T^{8} - 2125066040 T^{9} + 324961130 p T^{10} - 43187704 p^{2} T^{11} + 5495533 p^{3} T^{12} - 593867 p^{4} T^{13} + 61165 p^{5} T^{14} - 5020 p^{6} T^{15} + 392 p^{7} T^{16} - 20 p^{8} T^{17} + p^{9} T^{18} \)
43 \( 1 + 17 T + 227 T^{2} + 2140 T^{3} + 17914 T^{4} + 131075 T^{5} + 879969 T^{6} + 5320599 T^{7} + 32526199 T^{8} + 196873042 T^{9} + 32526199 p T^{10} + 5320599 p^{2} T^{11} + 879969 p^{3} T^{12} + 131075 p^{4} T^{13} + 17914 p^{5} T^{14} + 2140 p^{6} T^{15} + 227 p^{7} T^{16} + 17 p^{8} T^{17} + p^{9} T^{18} \)
47 \( 1 - 4 T + 147 T^{2} + 207 T^{3} + 6029 T^{4} + 68694 T^{5} + 197879 T^{6} + 3753473 T^{7} + 18796588 T^{8} + 136181628 T^{9} + 18796588 p T^{10} + 3753473 p^{2} T^{11} + 197879 p^{3} T^{12} + 68694 p^{4} T^{13} + 6029 p^{5} T^{14} + 207 p^{6} T^{15} + 147 p^{7} T^{16} - 4 p^{8} T^{17} + p^{9} T^{18} \)
53 \( 1 - 16 T + 317 T^{2} - 3576 T^{3} + 41298 T^{4} - 345801 T^{5} + 2982442 T^{6} - 20147904 T^{7} + 153700154 T^{8} - 1006557198 T^{9} + 153700154 p T^{10} - 20147904 p^{2} T^{11} + 2982442 p^{3} T^{12} - 345801 p^{4} T^{13} + 41298 p^{5} T^{14} - 3576 p^{6} T^{15} + 317 p^{7} T^{16} - 16 p^{8} T^{17} + p^{9} T^{18} \)
61 \( 1 + 9 T + 281 T^{2} + 3052 T^{3} + 49442 T^{4} + 467431 T^{5} + 5890869 T^{6} + 48018117 T^{7} + 488641059 T^{8} + 3485876686 T^{9} + 488641059 p T^{10} + 48018117 p^{2} T^{11} + 5890869 p^{3} T^{12} + 467431 p^{4} T^{13} + 49442 p^{5} T^{14} + 3052 p^{6} T^{15} + 281 p^{7} T^{16} + 9 p^{8} T^{17} + p^{9} T^{18} \)
67 \( 1 + 8 T + 361 T^{2} + 2209 T^{3} + 60341 T^{4} + 272426 T^{5} + 6379568 T^{6} + 21228770 T^{7} + 508519117 T^{8} + 1405021510 T^{9} + 508519117 p T^{10} + 21228770 p^{2} T^{11} + 6379568 p^{3} T^{12} + 272426 p^{4} T^{13} + 60341 p^{5} T^{14} + 2209 p^{6} T^{15} + 361 p^{7} T^{16} + 8 p^{8} T^{17} + p^{9} T^{18} \)
71 \( 1 + 8 T + 247 T^{2} + 1057 T^{3} + 25643 T^{4} - 14888 T^{5} + 1291597 T^{6} - 13800489 T^{7} + 31878792 T^{8} - 1460258384 T^{9} + 31878792 p T^{10} - 13800489 p^{2} T^{11} + 1291597 p^{3} T^{12} - 14888 p^{4} T^{13} + 25643 p^{5} T^{14} + 1057 p^{6} T^{15} + 247 p^{7} T^{16} + 8 p^{8} T^{17} + p^{9} T^{18} \)
73 \( 1 + 20 T + 534 T^{2} + 6762 T^{3} + 98191 T^{4} + 815125 T^{5} + 8049077 T^{6} + 39012798 T^{7} + 344322498 T^{8} + 1131087832 T^{9} + 344322498 p T^{10} + 39012798 p^{2} T^{11} + 8049077 p^{3} T^{12} + 815125 p^{4} T^{13} + 98191 p^{5} T^{14} + 6762 p^{6} T^{15} + 534 p^{7} T^{16} + 20 p^{8} T^{17} + p^{9} T^{18} \)
79 \( 1 + 29 T + 677 T^{2} + 11795 T^{3} + 182133 T^{4} + 2424297 T^{5} + 29519599 T^{6} + 322824554 T^{7} + 3261478800 T^{8} + 30087955658 T^{9} + 3261478800 p T^{10} + 322824554 p^{2} T^{11} + 29519599 p^{3} T^{12} + 2424297 p^{4} T^{13} + 182133 p^{5} T^{14} + 11795 p^{6} T^{15} + 677 p^{7} T^{16} + 29 p^{8} T^{17} + p^{9} T^{18} \)
83 \( 1 + 16 T + 560 T^{2} + 6742 T^{3} + 132344 T^{4} + 1305713 T^{5} + 18799240 T^{6} + 161145654 T^{7} + 1926967871 T^{8} + 14936632438 T^{9} + 1926967871 p T^{10} + 161145654 p^{2} T^{11} + 18799240 p^{3} T^{12} + 1305713 p^{4} T^{13} + 132344 p^{5} T^{14} + 6742 p^{6} T^{15} + 560 p^{7} T^{16} + 16 p^{8} T^{17} + p^{9} T^{18} \)
89 \( 1 - 11 T + 234 T^{2} - 2060 T^{3} + 33454 T^{4} - 239317 T^{5} + 3794919 T^{6} - 23446853 T^{7} + 360445499 T^{8} - 2272558860 T^{9} + 360445499 p T^{10} - 23446853 p^{2} T^{11} + 3794919 p^{3} T^{12} - 239317 p^{4} T^{13} + 33454 p^{5} T^{14} - 2060 p^{6} T^{15} + 234 p^{7} T^{16} - 11 p^{8} T^{17} + p^{9} T^{18} \)
97 \( 1 - 17 T + 356 T^{2} - 3429 T^{3} + 39554 T^{4} - 153349 T^{5} + 1037720 T^{6} + 18846449 T^{7} - 131487679 T^{8} + 3109904788 T^{9} - 131487679 p T^{10} + 18846449 p^{2} T^{11} + 1037720 p^{3} T^{12} - 153349 p^{4} T^{13} + 39554 p^{5} T^{14} - 3429 p^{6} T^{15} + 356 p^{7} T^{16} - 17 p^{8} T^{17} + p^{9} T^{18} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{18} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.30816358649020711506640847402, −3.20803595390808615238773362952, −3.04106535901092829190483917574, −2.95619305299139042830611124525, −2.92417586728736967896063380642, −2.90524072570435448266350345862, −2.75243229577905168254852450307, −2.32145082380181239535054629975, −2.27437897064725808886024733764, −2.26597162129341100991695390934, −2.24077112227464755253543514705, −2.14306842617079346435990071801, −2.11524561955648693979121330871, −2.04218422346585162635887690618, −2.01174788080385031461140839344, −1.86609125410200127584443712360, −1.40297907660403678351419025737, −1.38261318764869142319225984165, −1.34445680228574244214733624357, −1.28366019176610705564904658565, −1.17197778325730170127525979617, −1.14823963676105760798332385551, −1.03798301224389588950935381020, −0.871009945638245149291169476026, −0.815137783369469994570847349939, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.815137783369469994570847349939, 0.871009945638245149291169476026, 1.03798301224389588950935381020, 1.14823963676105760798332385551, 1.17197778325730170127525979617, 1.28366019176610705564904658565, 1.34445680228574244214733624357, 1.38261318764869142319225984165, 1.40297907660403678351419025737, 1.86609125410200127584443712360, 2.01174788080385031461140839344, 2.04218422346585162635887690618, 2.11524561955648693979121330871, 2.14306842617079346435990071801, 2.24077112227464755253543514705, 2.26597162129341100991695390934, 2.27437897064725808886024733764, 2.32145082380181239535054629975, 2.75243229577905168254852450307, 2.90524072570435448266350345862, 2.92417586728736967896063380642, 2.95619305299139042830611124525, 3.04106535901092829190483917574, 3.20803595390808615238773362952, 3.30816358649020711506640847402

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

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