Properties

Label 18-6042e9-1.1-c1e9-0-1
Degree $18$
Conductor $1.073\times 10^{34}$
Sign $1$
Analytic cond. $1.41618\times 10^{15}$
Root an. cond. $6.94590$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 9·2-s + 9·3-s + 45·4-s + 2·5-s − 81·6-s + 10·7-s − 165·8-s + 45·9-s − 18·10-s + 7·11-s + 405·12-s + 2·13-s − 90·14-s + 18·15-s + 495·16-s + 4·17-s − 405·18-s − 9·19-s + 90·20-s + 90·21-s − 63·22-s + 15·23-s − 1.48e3·24-s − 11·25-s − 18·26-s + 165·27-s + 450·28-s + ⋯
L(s)  = 1  − 6.36·2-s + 5.19·3-s + 45/2·4-s + 0.894·5-s − 33.0·6-s + 3.77·7-s − 58.3·8-s + 15·9-s − 5.69·10-s + 2.11·11-s + 116.·12-s + 0.554·13-s − 24.0·14-s + 4.64·15-s + 123.·16-s + 0.970·17-s − 95.4·18-s − 2.06·19-s + 20.1·20-s + 19.6·21-s − 13.4·22-s + 3.12·23-s − 303.·24-s − 2.19·25-s − 3.53·26-s + 31.7·27-s + 85.0·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{9} \cdot 19^{9} \cdot 53^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{9} \cdot 19^{9} \cdot 53^{9}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(18\)
Conductor: \(2^{9} \cdot 3^{9} \cdot 19^{9} \cdot 53^{9}\)
Sign: $1$
Analytic conductor: \(1.41618\times 10^{15}\)
Root analytic conductor: \(6.94590\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((18,\ 2^{9} \cdot 3^{9} \cdot 19^{9} \cdot 53^{9} ,\ ( \ : [1/2]^{9} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(209.4011055\)
\(L(\frac12)\) \(\approx\) \(209.4011055\)
\(L(\frac{3}{2})\) 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 \( ( 1 - T )^{9} \)
19 \( ( 1 + T )^{9} \)
53 \( ( 1 + T )^{9} \)
good5 \( 1 - 2 T + 3 p T^{2} - 23 T^{3} + 122 T^{4} - 91 T^{5} + 551 T^{6} + 253 T^{7} + 1711 T^{8} + 646 p T^{9} + 1711 p T^{10} + 253 p^{2} T^{11} + 551 p^{3} T^{12} - 91 p^{4} T^{13} + 122 p^{5} T^{14} - 23 p^{6} T^{15} + 3 p^{8} T^{16} - 2 p^{8} T^{17} + p^{9} T^{18} \)
7 \( 1 - 10 T + 65 T^{2} - 304 T^{3} + 1208 T^{4} - 4078 T^{5} + 12490 T^{6} - 34851 T^{7} + 94698 T^{8} - 249754 T^{9} + 94698 p T^{10} - 34851 p^{2} T^{11} + 12490 p^{3} T^{12} - 4078 p^{4} T^{13} + 1208 p^{5} T^{14} - 304 p^{6} T^{15} + 65 p^{7} T^{16} - 10 p^{8} T^{17} + p^{9} T^{18} \)
11 \( 1 - 7 T + 63 T^{2} - 371 T^{3} + 2204 T^{4} - 938 p T^{5} + 48308 T^{6} - 190661 T^{7} + 737962 T^{8} - 2472022 T^{9} + 737962 p T^{10} - 190661 p^{2} T^{11} + 48308 p^{3} T^{12} - 938 p^{5} T^{13} + 2204 p^{5} T^{14} - 371 p^{6} T^{15} + 63 p^{7} T^{16} - 7 p^{8} T^{17} + p^{9} T^{18} \)
13 \( 1 - 2 T + 67 T^{2} - 64 T^{3} + 1842 T^{4} + 1104 T^{5} + 26378 T^{6} + 6549 p T^{7} + 18932 p T^{8} + 1673402 T^{9} + 18932 p^{2} T^{10} + 6549 p^{3} T^{11} + 26378 p^{3} T^{12} + 1104 p^{4} T^{13} + 1842 p^{5} T^{14} - 64 p^{6} T^{15} + 67 p^{7} T^{16} - 2 p^{8} T^{17} + p^{9} T^{18} \)
17 \( 1 - 4 T + 95 T^{2} - 398 T^{3} + 4362 T^{4} - 18356 T^{5} + 129458 T^{6} - 31005 p T^{7} + 2821392 T^{8} - 10550450 T^{9} + 2821392 p T^{10} - 31005 p^{3} T^{11} + 129458 p^{3} T^{12} - 18356 p^{4} T^{13} + 4362 p^{5} T^{14} - 398 p^{6} T^{15} + 95 p^{7} T^{16} - 4 p^{8} T^{17} + p^{9} T^{18} \)
23 \( 1 - 15 T + 199 T^{2} - 1869 T^{3} + 16018 T^{4} - 115994 T^{5} + 773046 T^{6} - 4578867 T^{7} + 25217538 T^{8} - 125348622 T^{9} + 25217538 p T^{10} - 4578867 p^{2} T^{11} + 773046 p^{3} T^{12} - 115994 p^{4} T^{13} + 16018 p^{5} T^{14} - 1869 p^{6} T^{15} + 199 p^{7} T^{16} - 15 p^{8} T^{17} + p^{9} T^{18} \)
29 \( 1 - 5 T + 189 T^{2} - 26 p T^{3} + 16497 T^{4} - 54579 T^{5} + 905138 T^{6} - 2550651 T^{7} + 35227319 T^{8} - 85815094 T^{9} + 35227319 p T^{10} - 2550651 p^{2} T^{11} + 905138 p^{3} T^{12} - 54579 p^{4} T^{13} + 16497 p^{5} T^{14} - 26 p^{7} T^{15} + 189 p^{7} T^{16} - 5 p^{8} T^{17} + p^{9} T^{18} \)
31 \( 1 - T + 198 T^{2} - 142 T^{3} + 18928 T^{4} - 9904 T^{5} + 1149354 T^{6} - 461754 T^{7} + 48947231 T^{8} - 16199054 T^{9} + 48947231 p T^{10} - 461754 p^{2} T^{11} + 1149354 p^{3} T^{12} - 9904 p^{4} T^{13} + 18928 p^{5} T^{14} - 142 p^{6} T^{15} + 198 p^{7} T^{16} - p^{8} T^{17} + p^{9} T^{18} \)
37 \( 1 + 4 T + 195 T^{2} + 658 T^{3} + 19960 T^{4} + 58410 T^{5} + 1356172 T^{6} + 3440965 T^{7} + 66883884 T^{8} + 148089038 T^{9} + 66883884 p T^{10} + 3440965 p^{2} T^{11} + 1356172 p^{3} T^{12} + 58410 p^{4} T^{13} + 19960 p^{5} T^{14} + 658 p^{6} T^{15} + 195 p^{7} T^{16} + 4 p^{8} T^{17} + p^{9} T^{18} \)
41 \( 1 - 5 T + 315 T^{2} - 1369 T^{3} + 45208 T^{4} - 170362 T^{5} + 3922507 T^{6} - 12739029 T^{7} + 229032721 T^{8} - 632291654 T^{9} + 229032721 p T^{10} - 12739029 p^{2} T^{11} + 3922507 p^{3} T^{12} - 170362 p^{4} T^{13} + 45208 p^{5} T^{14} - 1369 p^{6} T^{15} + 315 p^{7} T^{16} - 5 p^{8} T^{17} + p^{9} T^{18} \)
43 \( 1 - 27 T + 525 T^{2} - 7591 T^{3} + 94556 T^{4} - 1001202 T^{5} + 9482608 T^{6} - 79558113 T^{7} + 607521420 T^{8} - 4168110150 T^{9} + 607521420 p T^{10} - 79558113 p^{2} T^{11} + 9482608 p^{3} T^{12} - 1001202 p^{4} T^{13} + 94556 p^{5} T^{14} - 7591 p^{6} T^{15} + 525 p^{7} T^{16} - 27 p^{8} T^{17} + p^{9} T^{18} \)
47 \( 1 + 252 T^{2} + 301 T^{3} + 31382 T^{4} + 56410 T^{5} + 2632137 T^{6} + 4987774 T^{7} + 163353922 T^{8} + 282222870 T^{9} + 163353922 p T^{10} + 4987774 p^{2} T^{11} + 2632137 p^{3} T^{12} + 56410 p^{4} T^{13} + 31382 p^{5} T^{14} + 301 p^{6} T^{15} + 252 p^{7} T^{16} + p^{9} T^{18} \)
59 \( 1 - 7 T + 151 T^{2} - 120 T^{3} + 8685 T^{4} + 32869 T^{5} + 617438 T^{6} + 3000109 T^{7} + 35453595 T^{8} + 258722514 T^{9} + 35453595 p T^{10} + 3000109 p^{2} T^{11} + 617438 p^{3} T^{12} + 32869 p^{4} T^{13} + 8685 p^{5} T^{14} - 120 p^{6} T^{15} + 151 p^{7} T^{16} - 7 p^{8} T^{17} + p^{9} T^{18} \)
61 \( 1 - 5 T + 351 T^{2} - 1470 T^{3} + 60893 T^{4} - 222337 T^{5} + 6879394 T^{6} - 22133155 T^{7} + 560335029 T^{8} - 1580330634 T^{9} + 560335029 p T^{10} - 22133155 p^{2} T^{11} + 6879394 p^{3} T^{12} - 222337 p^{4} T^{13} + 60893 p^{5} T^{14} - 1470 p^{6} T^{15} + 351 p^{7} T^{16} - 5 p^{8} T^{17} + p^{9} T^{18} \)
67 \( 1 - 19 T + 607 T^{2} - 8649 T^{3} + 159722 T^{4} - 1814070 T^{5} + 24462390 T^{6} - 227645407 T^{7} + 2428480718 T^{8} - 18655767246 T^{9} + 2428480718 p T^{10} - 227645407 p^{2} T^{11} + 24462390 p^{3} T^{12} - 1814070 p^{4} T^{13} + 159722 p^{5} T^{14} - 8649 p^{6} T^{15} + 607 p^{7} T^{16} - 19 p^{8} T^{17} + p^{9} T^{18} \)
71 \( 1 + 11 T + 334 T^{2} + 2758 T^{3} + 53246 T^{4} + 388336 T^{5} + 5998412 T^{6} + 41576578 T^{7} + 541707531 T^{8} + 3442504058 T^{9} + 541707531 p T^{10} + 41576578 p^{2} T^{11} + 5998412 p^{3} T^{12} + 388336 p^{4} T^{13} + 53246 p^{5} T^{14} + 2758 p^{6} T^{15} + 334 p^{7} T^{16} + 11 p^{8} T^{17} + p^{9} T^{18} \)
73 \( 1 - 25 T + 617 T^{2} - 133 p T^{3} + 145754 T^{4} - 1774610 T^{5} + 20605810 T^{6} - 210833539 T^{7} + 2047487906 T^{8} - 17992552746 T^{9} + 2047487906 p T^{10} - 210833539 p^{2} T^{11} + 20605810 p^{3} T^{12} - 1774610 p^{4} T^{13} + 145754 p^{5} T^{14} - 133 p^{7} T^{15} + 617 p^{7} T^{16} - 25 p^{8} T^{17} + p^{9} T^{18} \)
79 \( 1 - 28 T + 697 T^{2} - 10703 T^{3} + 152712 T^{4} - 1648163 T^{5} + 17604103 T^{6} - 155693933 T^{7} + 1480820123 T^{8} - 12396314634 T^{9} + 1480820123 p T^{10} - 155693933 p^{2} T^{11} + 17604103 p^{3} T^{12} - 1648163 p^{4} T^{13} + 152712 p^{5} T^{14} - 10703 p^{6} T^{15} + 697 p^{7} T^{16} - 28 p^{8} T^{17} + p^{9} T^{18} \)
83 \( 1 + 8 T + 221 T^{2} + 1226 T^{3} + 32318 T^{4} + 143636 T^{5} + 2982958 T^{6} + 6129043 T^{7} + 237314108 T^{8} + 506463718 T^{9} + 237314108 p T^{10} + 6129043 p^{2} T^{11} + 2982958 p^{3} T^{12} + 143636 p^{4} T^{13} + 32318 p^{5} T^{14} + 1226 p^{6} T^{15} + 221 p^{7} T^{16} + 8 p^{8} T^{17} + p^{9} T^{18} \)
89 \( 1 - 22 T + 616 T^{2} - 8235 T^{3} + 134509 T^{4} - 1316330 T^{5} + 17534367 T^{6} - 149126805 T^{7} + 1864092435 T^{8} - 14564305056 T^{9} + 1864092435 p T^{10} - 149126805 p^{2} T^{11} + 17534367 p^{3} T^{12} - 1316330 p^{4} T^{13} + 134509 p^{5} T^{14} - 8235 p^{6} T^{15} + 616 p^{7} T^{16} - 22 p^{8} T^{17} + p^{9} T^{18} \)
97 \( 1 - 13 T + 685 T^{2} - 7967 T^{3} + 218294 T^{4} - 2254626 T^{5} + 42954494 T^{6} - 389449057 T^{7} + 5809329214 T^{8} - 45366455586 T^{9} + 5809329214 p T^{10} - 389449057 p^{2} T^{11} + 42954494 p^{3} T^{12} - 2254626 p^{4} T^{13} + 218294 p^{5} T^{14} - 7967 p^{6} T^{15} + 685 p^{7} T^{16} - 13 p^{8} T^{17} + p^{9} T^{18} \)
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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

−2.78533508606323705306474226731, −2.74201162559711992060264208702, −2.55889602442745927791684266468, −2.53334282614749085799585107100, −2.43851631665970976588695036919, −2.33744000626422324729040813817, −2.28131032891735318285270587119, −1.99419960986636153246292206290, −1.97801118839337479288843931053, −1.95817261479563760572294462432, −1.83200501106103824015818184691, −1.82719573087102497607771249493, −1.75601729709579697897644959746, −1.73507094458818805560786276494, −1.69465609139337723278176604355, −1.48768202244720110125362376606, −1.21090718255665752319530673002, −1.17195893290881233443644508540, −1.04137758671867455270223509537, −0.941824144031553843319637999226, −0.76469207649368739770570208142, −0.70759614321334485880884015468, −0.64056609324540604316253402698, −0.63041395774518832466210163997, −0.43681590898931258232636267582, 0.43681590898931258232636267582, 0.63041395774518832466210163997, 0.64056609324540604316253402698, 0.70759614321334485880884015468, 0.76469207649368739770570208142, 0.941824144031553843319637999226, 1.04137758671867455270223509537, 1.17195893290881233443644508540, 1.21090718255665752319530673002, 1.48768202244720110125362376606, 1.69465609139337723278176604355, 1.73507094458818805560786276494, 1.75601729709579697897644959746, 1.82719573087102497607771249493, 1.83200501106103824015818184691, 1.95817261479563760572294462432, 1.97801118839337479288843931053, 1.99419960986636153246292206290, 2.28131032891735318285270587119, 2.33744000626422324729040813817, 2.43851631665970976588695036919, 2.53334282614749085799585107100, 2.55889602442745927791684266468, 2.74201162559711992060264208702, 2.78533508606323705306474226731

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

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