Properties

Label 18-3380e9-1.1-c1e9-0-1
Degree $18$
Conductor $5.758\times 10^{31}$
Sign $1$
Analytic cond. $7.59878\times 10^{12}$
Root an. cond. $5.19513$
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  − 3-s + 9·5-s + 7-s − 7·9-s − 7·11-s − 9·15-s + 13·17-s − 4·19-s − 21-s + 12·23-s + 45·25-s + 7·27-s + 16·29-s + 13·31-s + 7·33-s + 9·35-s + 37-s − 6·41-s + 43-s − 63·45-s − 2·47-s − 21·49-s − 13·51-s + 30·53-s − 63·55-s + 4·57-s + 15·59-s + ⋯
L(s)  = 1  − 0.577·3-s + 4.02·5-s + 0.377·7-s − 7/3·9-s − 2.11·11-s − 2.32·15-s + 3.15·17-s − 0.917·19-s − 0.218·21-s + 2.50·23-s + 9·25-s + 1.34·27-s + 2.97·29-s + 2.33·31-s + 1.21·33-s + 1.52·35-s + 0.164·37-s − 0.937·41-s + 0.152·43-s − 9.39·45-s − 0.291·47-s − 3·49-s − 1.82·51-s + 4.12·53-s − 8.49·55-s + 0.529·57-s + 1.95·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{9} \cdot 13^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{9} \cdot 13^{18}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(18\)
Conductor: \(2^{18} \cdot 5^{9} \cdot 13^{18}\)
Sign: $1$
Analytic conductor: \(7.59878\times 10^{12}\)
Root analytic conductor: \(5.19513\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((18,\ 2^{18} \cdot 5^{9} \cdot 13^{18} ,\ ( \ : [1/2]^{9} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(68.90155144\)
\(L(\frac12)\) \(\approx\) \(68.90155144\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( ( 1 - T )^{9} \)
13 \( 1 \)
good3 \( 1 + T + 8 T^{2} + 8 T^{3} + 31 T^{4} + 17 p T^{5} + 38 p T^{6} + 247 T^{7} + 388 T^{8} + 833 T^{9} + 388 p T^{10} + 247 p^{2} T^{11} + 38 p^{4} T^{12} + 17 p^{5} T^{13} + 31 p^{5} T^{14} + 8 p^{6} T^{15} + 8 p^{7} T^{16} + p^{8} T^{17} + p^{9} T^{18} \)
7 \( 1 - T + 22 T^{2} + 4 T^{3} + 39 p T^{4} + 197 T^{5} + 2876 T^{6} + 2173 T^{7} + 25850 T^{8} + 15541 T^{9} + 25850 p T^{10} + 2173 p^{2} T^{11} + 2876 p^{3} T^{12} + 197 p^{4} T^{13} + 39 p^{6} T^{14} + 4 p^{6} T^{15} + 22 p^{7} T^{16} - p^{8} T^{17} + p^{9} T^{18} \)
11 \( 1 + 7 T + 41 T^{2} + 13 p T^{3} + 582 T^{4} + 2490 T^{5} + 11926 T^{6} + 43145 T^{7} + 145500 T^{8} + 428286 T^{9} + 145500 p T^{10} + 43145 p^{2} T^{11} + 11926 p^{3} T^{12} + 2490 p^{4} T^{13} + 582 p^{5} T^{14} + 13 p^{7} T^{15} + 41 p^{7} T^{16} + 7 p^{8} T^{17} + p^{9} T^{18} \)
17 \( 1 - 13 T + 127 T^{2} - 891 T^{3} + 5818 T^{4} - 33238 T^{5} + 180078 T^{6} - 879477 T^{7} + 4060136 T^{8} - 17187546 T^{9} + 4060136 p T^{10} - 879477 p^{2} T^{11} + 180078 p^{3} T^{12} - 33238 p^{4} T^{13} + 5818 p^{5} T^{14} - 891 p^{6} T^{15} + 127 p^{7} T^{16} - 13 p^{8} T^{17} + p^{9} T^{18} \)
19 \( 1 + 4 T + 84 T^{2} + 355 T^{3} + 3541 T^{4} + 15002 T^{5} + 110061 T^{6} + 429333 T^{7} + 2741855 T^{8} + 9310468 T^{9} + 2741855 p T^{10} + 429333 p^{2} T^{11} + 110061 p^{3} T^{12} + 15002 p^{4} T^{13} + 3541 p^{5} T^{14} + 355 p^{6} T^{15} + 84 p^{7} T^{16} + 4 p^{8} T^{17} + p^{9} T^{18} \)
23 \( 1 - 12 T + 173 T^{2} - 1335 T^{3} + 10807 T^{4} - 61633 T^{5} + 359296 T^{6} - 1685957 T^{7} + 8393430 T^{8} - 38069843 T^{9} + 8393430 p T^{10} - 1685957 p^{2} T^{11} + 359296 p^{3} T^{12} - 61633 p^{4} T^{13} + 10807 p^{5} T^{14} - 1335 p^{6} T^{15} + 173 p^{7} T^{16} - 12 p^{8} T^{17} + p^{9} T^{18} \)
29 \( 1 - 16 T + 219 T^{2} - 2133 T^{3} + 18623 T^{4} - 134367 T^{5} + 906672 T^{6} - 5432131 T^{7} + 31678126 T^{8} - 170565345 T^{9} + 31678126 p T^{10} - 5432131 p^{2} T^{11} + 906672 p^{3} T^{12} - 134367 p^{4} T^{13} + 18623 p^{5} T^{14} - 2133 p^{6} T^{15} + 219 p^{7} T^{16} - 16 p^{8} T^{17} + p^{9} T^{18} \)
31 \( 1 - 13 T + 167 T^{2} - 1499 T^{3} + 13436 T^{4} - 95962 T^{5} + 698532 T^{6} - 4410373 T^{7} + 27899758 T^{8} - 154411698 T^{9} + 27899758 p T^{10} - 4410373 p^{2} T^{11} + 698532 p^{3} T^{12} - 95962 p^{4} T^{13} + 13436 p^{5} T^{14} - 1499 p^{6} T^{15} + 167 p^{7} T^{16} - 13 p^{8} T^{17} + p^{9} T^{18} \)
37 \( 1 - T + 164 T^{2} + 25 T^{3} + 13345 T^{4} + 19962 T^{5} + 748055 T^{6} + 1820111 T^{7} + 33023343 T^{8} + 86331982 T^{9} + 33023343 p T^{10} + 1820111 p^{2} T^{11} + 748055 p^{3} T^{12} + 19962 p^{4} T^{13} + 13345 p^{5} T^{14} + 25 p^{6} T^{15} + 164 p^{7} T^{16} - p^{8} T^{17} + p^{9} T^{18} \)
41 \( 1 + 6 T + 105 T^{2} + 657 T^{3} + 8491 T^{4} + 56617 T^{5} + 512164 T^{6} + 3237269 T^{7} + 24968146 T^{8} + 155625947 T^{9} + 24968146 p T^{10} + 3237269 p^{2} T^{11} + 512164 p^{3} T^{12} + 56617 p^{4} T^{13} + 8491 p^{5} T^{14} + 657 p^{6} T^{15} + 105 p^{7} T^{16} + 6 p^{8} T^{17} + p^{9} T^{18} \)
43 \( 1 - T + 144 T^{2} + 36 T^{3} + 14051 T^{4} + 4699 T^{5} + 997254 T^{6} + 464435 T^{7} + 54046602 T^{8} + 25642497 T^{9} + 54046602 p T^{10} + 464435 p^{2} T^{11} + 997254 p^{3} T^{12} + 4699 p^{4} T^{13} + 14051 p^{5} T^{14} + 36 p^{6} T^{15} + 144 p^{7} T^{16} - p^{8} T^{17} + p^{9} T^{18} \)
47 \( 1 + 2 T + 157 T^{2} + 27 T^{3} + 15847 T^{4} + 2755 T^{5} + 1203316 T^{6} - 22677 T^{7} + 70019046 T^{8} - 2721691 T^{9} + 70019046 p T^{10} - 22677 p^{2} T^{11} + 1203316 p^{3} T^{12} + 2755 p^{4} T^{13} + 15847 p^{5} T^{14} + 27 p^{6} T^{15} + 157 p^{7} T^{16} + 2 p^{8} T^{17} + p^{9} T^{18} \)
53 \( 1 - 30 T + 764 T^{2} - 13045 T^{3} + 197201 T^{4} - 2405102 T^{5} + 26608679 T^{6} - 251697171 T^{7} + 2186503079 T^{8} - 16578690952 T^{9} + 2186503079 p T^{10} - 251697171 p^{2} T^{11} + 26608679 p^{3} T^{12} - 2405102 p^{4} T^{13} + 197201 p^{5} T^{14} - 13045 p^{6} T^{15} + 764 p^{7} T^{16} - 30 p^{8} T^{17} + p^{9} T^{18} \)
59 \( 1 - 15 T + 410 T^{2} - 4305 T^{3} + 65047 T^{4} - 487110 T^{5} + 5390959 T^{6} - 29366503 T^{7} + 302432813 T^{8} - 1478780406 T^{9} + 302432813 p T^{10} - 29366503 p^{2} T^{11} + 5390959 p^{3} T^{12} - 487110 p^{4} T^{13} + 65047 p^{5} T^{14} - 4305 p^{6} T^{15} + 410 p^{7} T^{16} - 15 p^{8} T^{17} + p^{9} T^{18} \)
61 \( 1 - 21 T + 450 T^{2} - 6360 T^{3} + 87635 T^{4} - 952857 T^{5} + 10212800 T^{6} - 92578873 T^{7} + 832837852 T^{8} - 6507880653 T^{9} + 832837852 p T^{10} - 92578873 p^{2} T^{11} + 10212800 p^{3} T^{12} - 952857 p^{4} T^{13} + 87635 p^{5} T^{14} - 6360 p^{6} T^{15} + 450 p^{7} T^{16} - 21 p^{8} T^{17} + p^{9} T^{18} \)
67 \( 1 + 7 T + 271 T^{2} + 862 T^{3} + 31617 T^{4} + 22471 T^{5} + 2747942 T^{6} - 340663 T^{7} + 220931030 T^{8} - 6142161 T^{9} + 220931030 p T^{10} - 340663 p^{2} T^{11} + 2747942 p^{3} T^{12} + 22471 p^{4} T^{13} + 31617 p^{5} T^{14} + 862 p^{6} T^{15} + 271 p^{7} T^{16} + 7 p^{8} T^{17} + p^{9} T^{18} \)
71 \( 1 + 7 T + 329 T^{2} + 1655 T^{3} + 57246 T^{4} + 227682 T^{5} + 6713342 T^{6} + 20970873 T^{7} + 596832992 T^{8} + 1621578350 T^{9} + 596832992 p T^{10} + 20970873 p^{2} T^{11} + 6713342 p^{3} T^{12} + 227682 p^{4} T^{13} + 57246 p^{5} T^{14} + 1655 p^{6} T^{15} + 329 p^{7} T^{16} + 7 p^{8} T^{17} + p^{9} T^{18} \)
73 \( 1 + 28 T + 593 T^{2} + 9272 T^{3} + 133028 T^{4} + 1626944 T^{5} + 18735348 T^{6} + 192384968 T^{7} + 1868388734 T^{8} + 16364819272 T^{9} + 1868388734 p T^{10} + 192384968 p^{2} T^{11} + 18735348 p^{3} T^{12} + 1626944 p^{4} T^{13} + 133028 p^{5} T^{14} + 9272 p^{6} T^{15} + 593 p^{7} T^{16} + 28 p^{8} T^{17} + p^{9} T^{18} \)
79 \( 1 - 31 T + 757 T^{2} - 12457 T^{3} + 188314 T^{4} - 2369594 T^{5} + 28747466 T^{6} - 304054231 T^{7} + 3100351316 T^{8} - 27989535950 T^{9} + 3100351316 p T^{10} - 304054231 p^{2} T^{11} + 28747466 p^{3} T^{12} - 2369594 p^{4} T^{13} + 188314 p^{5} T^{14} - 12457 p^{6} T^{15} + 757 p^{7} T^{16} - 31 p^{8} T^{17} + p^{9} T^{18} \)
83 \( 1 - 45 T + 1348 T^{2} - 29160 T^{3} + 518079 T^{4} - 7760711 T^{5} + 102182454 T^{6} - 1194959175 T^{7} + 12622802288 T^{8} - 120577066249 T^{9} + 12622802288 p T^{10} - 1194959175 p^{2} T^{11} + 102182454 p^{3} T^{12} - 7760711 p^{4} T^{13} + 518079 p^{5} T^{14} - 29160 p^{6} T^{15} + 1348 p^{7} T^{16} - 45 p^{8} T^{17} + p^{9} T^{18} \)
89 \( 1 + 41 T + 1239 T^{2} + 26092 T^{3} + 471499 T^{4} + 7079921 T^{5} + 96248956 T^{6} + 1149115833 T^{7} + 12643775152 T^{8} + 124080739275 T^{9} + 12643775152 p T^{10} + 1149115833 p^{2} T^{11} + 96248956 p^{3} T^{12} + 7079921 p^{4} T^{13} + 471499 p^{5} T^{14} + 26092 p^{6} T^{15} + 1239 p^{7} T^{16} + 41 p^{8} T^{17} + p^{9} T^{18} \)
97 \( 1 - 8 T + 472 T^{2} - 4349 T^{3} + 122457 T^{4} - 1088022 T^{5} + 21739787 T^{6} - 174776115 T^{7} + 2806532707 T^{8} - 19946280100 T^{9} + 2806532707 p T^{10} - 174776115 p^{2} T^{11} + 21739787 p^{3} T^{12} - 1088022 p^{4} T^{13} + 122457 p^{5} T^{14} - 4349 p^{6} T^{15} + 472 p^{7} T^{16} - 8 p^{8} T^{17} + p^{9} T^{18} \)
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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

−3.07910391237843663535014692760, −3.00336346494981775078377321833, −2.99132464059165244383789828234, −2.98702782145413618243323985569, −2.95007637433808695082237977599, −2.75376957535432526209444905846, −2.72141173595838055066811673341, −2.41443652197356631218872023424, −2.34791729730145513620628531341, −2.29057776353970211763642506790, −2.11223504250768218648774329743, −2.02591443957035818811741365489, −1.90464704981872809978204965418, −1.89317159041907190155873718855, −1.88670388177879760212988506181, −1.67621289329591754855738142511, −1.34339344013668806985292165979, −1.12699996148970207141020280584, −1.04633069803559447825296680710, −0.991377634261608924746820643259, −0.844509090418275146249379088038, −0.813214629514160300533718717426, −0.68493147320691207814328436796, −0.39141242794799086753314643804, −0.36434352815240505160325369029, 0.36434352815240505160325369029, 0.39141242794799086753314643804, 0.68493147320691207814328436796, 0.813214629514160300533718717426, 0.844509090418275146249379088038, 0.991377634261608924746820643259, 1.04633069803559447825296680710, 1.12699996148970207141020280584, 1.34339344013668806985292165979, 1.67621289329591754855738142511, 1.88670388177879760212988506181, 1.89317159041907190155873718855, 1.90464704981872809978204965418, 2.02591443957035818811741365489, 2.11223504250768218648774329743, 2.29057776353970211763642506790, 2.34791729730145513620628531341, 2.41443652197356631218872023424, 2.72141173595838055066811673341, 2.75376957535432526209444905846, 2.95007637433808695082237977599, 2.98702782145413618243323985569, 2.99132464059165244383789828234, 3.00336346494981775078377321833, 3.07910391237843663535014692760

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

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