Properties

Label 18-8016e9-1.1-c1e9-0-0
Degree $18$
Conductor $1.367\times 10^{35}$
Sign $1$
Analytic cond. $1.80348\times 10^{16}$
Root an. cond. $8.00050$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 9·3-s + 9·5-s − 2·7-s + 45·9-s − 7·11-s + 6·13-s − 81·15-s + 7·17-s − 2·19-s + 18·21-s − 19·23-s + 29·25-s − 165·27-s + 13·29-s − 12·31-s + 63·33-s − 18·35-s + 15·37-s − 54·39-s + 18·41-s + 6·43-s + 405·45-s − 25·47-s − 20·49-s − 63·51-s + 17·53-s − 63·55-s + ⋯
L(s)  = 1  − 5.19·3-s + 4.02·5-s − 0.755·7-s + 15·9-s − 2.11·11-s + 1.66·13-s − 20.9·15-s + 1.69·17-s − 0.458·19-s + 3.92·21-s − 3.96·23-s + 29/5·25-s − 31.7·27-s + 2.41·29-s − 2.15·31-s + 10.9·33-s − 3.04·35-s + 2.46·37-s − 8.64·39-s + 2.81·41-s + 0.914·43-s + 60.3·45-s − 3.64·47-s − 2.85·49-s − 8.82·51-s + 2.33·53-s − 8.49·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{9} \cdot 167^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{9} \cdot 167^{9}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(18\)
Conductor: \(2^{36} \cdot 3^{9} \cdot 167^{9}\)
Sign: $1$
Analytic conductor: \(1.80348\times 10^{16}\)
Root analytic conductor: \(8.00050\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{8016} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((18,\ 2^{36} \cdot 3^{9} \cdot 167^{9} ,\ ( \ : [1/2]^{9} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.7879867307\)
\(L(\frac12)\) \(\approx\) \(0.7879867307\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( ( 1 + T )^{9} \)
167 \( ( 1 + T )^{9} \)
good5 \( 1 - 9 T + 52 T^{2} - 234 T^{3} + 886 T^{4} - 117 p^{2} T^{5} + 8664 T^{6} - 23344 T^{7} + 58229 T^{8} - 134906 T^{9} + 58229 p T^{10} - 23344 p^{2} T^{11} + 8664 p^{3} T^{12} - 117 p^{6} T^{13} + 886 p^{5} T^{14} - 234 p^{6} T^{15} + 52 p^{7} T^{16} - 9 p^{8} T^{17} + p^{9} T^{18} \)
7 \( 1 + 2 T + 24 T^{2} + 3 T^{3} + 204 T^{4} - 67 p T^{5} + 1446 T^{6} - 5119 T^{7} + 15489 T^{8} - 32962 T^{9} + 15489 p T^{10} - 5119 p^{2} T^{11} + 1446 p^{3} T^{12} - 67 p^{5} T^{13} + 204 p^{5} T^{14} + 3 p^{6} T^{15} + 24 p^{7} T^{16} + 2 p^{8} T^{17} + p^{9} T^{18} \)
11 \( 1 + 7 T + 76 T^{2} + 448 T^{3} + 2798 T^{4} + 13617 T^{5} + 64556 T^{6} + 259240 T^{7} + 1013497 T^{8} + 3395532 T^{9} + 1013497 p T^{10} + 259240 p^{2} T^{11} + 64556 p^{3} T^{12} + 13617 p^{4} T^{13} + 2798 p^{5} T^{14} + 448 p^{6} T^{15} + 76 p^{7} T^{16} + 7 p^{8} T^{17} + p^{9} T^{18} \)
13 \( 1 - 6 T + 72 T^{2} - 290 T^{3} + 2079 T^{4} - 5583 T^{5} + 2501 p T^{6} - 50538 T^{7} + 355779 T^{8} - 363982 T^{9} + 355779 p T^{10} - 50538 p^{2} T^{11} + 2501 p^{4} T^{12} - 5583 p^{4} T^{13} + 2079 p^{5} T^{14} - 290 p^{6} T^{15} + 72 p^{7} T^{16} - 6 p^{8} T^{17} + p^{9} T^{18} \)
17 \( 1 - 7 T + 95 T^{2} - 412 T^{3} + 3239 T^{4} - 7217 T^{5} + 46207 T^{6} + 42146 T^{7} + 182922 T^{8} + 2723810 T^{9} + 182922 p T^{10} + 42146 p^{2} T^{11} + 46207 p^{3} T^{12} - 7217 p^{4} T^{13} + 3239 p^{5} T^{14} - 412 p^{6} T^{15} + 95 p^{7} T^{16} - 7 p^{8} T^{17} + p^{9} T^{18} \)
19 \( 1 + 2 T + 102 T^{2} + 170 T^{3} + 237 p T^{4} + 4717 T^{5} + 117347 T^{6} + 26098 T^{7} + 2285365 T^{8} - 603238 T^{9} + 2285365 p T^{10} + 26098 p^{2} T^{11} + 117347 p^{3} T^{12} + 4717 p^{4} T^{13} + 237 p^{6} T^{14} + 170 p^{6} T^{15} + 102 p^{7} T^{16} + 2 p^{8} T^{17} + p^{9} T^{18} \)
23 \( 1 + 19 T + 254 T^{2} + 2284 T^{3} + 17484 T^{4} + 110157 T^{5} + 659522 T^{6} + 3574832 T^{7} + 19162755 T^{8} + 4049112 p T^{9} + 19162755 p T^{10} + 3574832 p^{2} T^{11} + 659522 p^{3} T^{12} + 110157 p^{4} T^{13} + 17484 p^{5} T^{14} + 2284 p^{6} T^{15} + 254 p^{7} T^{16} + 19 p^{8} T^{17} + p^{9} T^{18} \)
29 \( 1 - 13 T + 154 T^{2} - 1330 T^{3} + 10268 T^{4} - 65265 T^{5} + 395422 T^{6} - 2088442 T^{7} + 11323883 T^{8} - 58112060 T^{9} + 11323883 p T^{10} - 2088442 p^{2} T^{11} + 395422 p^{3} T^{12} - 65265 p^{4} T^{13} + 10268 p^{5} T^{14} - 1330 p^{6} T^{15} + 154 p^{7} T^{16} - 13 p^{8} T^{17} + p^{9} T^{18} \)
31 \( 1 + 12 T + 237 T^{2} + 2081 T^{3} + 24736 T^{4} + 175357 T^{5} + 1571692 T^{6} + 9333683 T^{7} + 68244056 T^{8} + 343757830 T^{9} + 68244056 p T^{10} + 9333683 p^{2} T^{11} + 1571692 p^{3} T^{12} + 175357 p^{4} T^{13} + 24736 p^{5} T^{14} + 2081 p^{6} T^{15} + 237 p^{7} T^{16} + 12 p^{8} T^{17} + p^{9} T^{18} \)
37 \( 1 - 15 T + 242 T^{2} - 2138 T^{3} + 21014 T^{4} - 145273 T^{5} + 1219856 T^{6} - 212410 p T^{7} + 60067423 T^{8} - 344194216 T^{9} + 60067423 p T^{10} - 212410 p^{3} T^{11} + 1219856 p^{3} T^{12} - 145273 p^{4} T^{13} + 21014 p^{5} T^{14} - 2138 p^{6} T^{15} + 242 p^{7} T^{16} - 15 p^{8} T^{17} + p^{9} T^{18} \)
41 \( 1 - 18 T + 391 T^{2} - 4825 T^{3} + 61116 T^{4} - 575535 T^{5} + 5406478 T^{6} - 41258085 T^{7} + 314515424 T^{8} - 2011563376 T^{9} + 314515424 p T^{10} - 41258085 p^{2} T^{11} + 5406478 p^{3} T^{12} - 575535 p^{4} T^{13} + 61116 p^{5} T^{14} - 4825 p^{6} T^{15} + 391 p^{7} T^{16} - 18 p^{8} T^{17} + p^{9} T^{18} \)
43 \( 1 - 6 T + 97 T^{2} - 675 T^{3} + 5484 T^{4} - 22849 T^{5} + 166420 T^{6} - 187403 T^{7} + 1595534 T^{8} + 10660100 T^{9} + 1595534 p T^{10} - 187403 p^{2} T^{11} + 166420 p^{3} T^{12} - 22849 p^{4} T^{13} + 5484 p^{5} T^{14} - 675 p^{6} T^{15} + 97 p^{7} T^{16} - 6 p^{8} T^{17} + p^{9} T^{18} \)
47 \( 1 + 25 T + 448 T^{2} + 121 p T^{3} + 1367 p T^{4} + 630769 T^{5} + 5789701 T^{6} + 47606857 T^{7} + 365856543 T^{8} + 2574158528 T^{9} + 365856543 p T^{10} + 47606857 p^{2} T^{11} + 5789701 p^{3} T^{12} + 630769 p^{4} T^{13} + 1367 p^{6} T^{14} + 121 p^{7} T^{15} + 448 p^{7} T^{16} + 25 p^{8} T^{17} + p^{9} T^{18} \)
53 \( 1 - 17 T + 6 p T^{2} - 3079 T^{3} + 33015 T^{4} - 222369 T^{5} + 1931189 T^{6} - 10783739 T^{7} + 96671639 T^{8} - 529914954 T^{9} + 96671639 p T^{10} - 10783739 p^{2} T^{11} + 1931189 p^{3} T^{12} - 222369 p^{4} T^{13} + 33015 p^{5} T^{14} - 3079 p^{6} T^{15} + 6 p^{8} T^{16} - 17 p^{8} T^{17} + p^{9} T^{18} \)
59 \( 1 + 3 T + 248 T^{2} + 587 T^{3} + 35733 T^{4} + 67161 T^{5} + 3580941 T^{6} + 97955 p T^{7} + 272819099 T^{8} + 381917456 T^{9} + 272819099 p T^{10} + 97955 p^{3} T^{11} + 3580941 p^{3} T^{12} + 67161 p^{4} T^{13} + 35733 p^{5} T^{14} + 587 p^{6} T^{15} + 248 p^{7} T^{16} + 3 p^{8} T^{17} + p^{9} T^{18} \)
61 \( 1 - 14 T + 286 T^{2} - 2662 T^{3} + 34623 T^{4} - 221617 T^{5} + 2297933 T^{6} - 9978522 T^{7} + 111413285 T^{8} - 393019054 T^{9} + 111413285 p T^{10} - 9978522 p^{2} T^{11} + 2297933 p^{3} T^{12} - 221617 p^{4} T^{13} + 34623 p^{5} T^{14} - 2662 p^{6} T^{15} + 286 p^{7} T^{16} - 14 p^{8} T^{17} + p^{9} T^{18} \)
67 \( 1 - 4 T + 6 p T^{2} - 1172 T^{3} + 77435 T^{4} - 162163 T^{5} + 9543569 T^{6} - 14628924 T^{7} + 845476133 T^{8} - 1048306460 T^{9} + 845476133 p T^{10} - 14628924 p^{2} T^{11} + 9543569 p^{3} T^{12} - 162163 p^{4} T^{13} + 77435 p^{5} T^{14} - 1172 p^{6} T^{15} + 6 p^{8} T^{16} - 4 p^{8} T^{17} + p^{9} T^{18} \)
71 \( 1 + 17 T + 319 T^{2} + 2841 T^{3} + 39999 T^{4} + 355099 T^{5} + 69615 p T^{6} + 38075859 T^{7} + 407671928 T^{8} + 2653132208 T^{9} + 407671928 p T^{10} + 38075859 p^{2} T^{11} + 69615 p^{4} T^{12} + 355099 p^{4} T^{13} + 39999 p^{5} T^{14} + 2841 p^{6} T^{15} + 319 p^{7} T^{16} + 17 p^{8} T^{17} + p^{9} T^{18} \)
73 \( 1 + 20 T + 498 T^{2} + 7086 T^{3} + 112237 T^{4} + 1284803 T^{5} + 15841175 T^{6} + 152923942 T^{7} + 1573749217 T^{8} + 13055799498 T^{9} + 1573749217 p T^{10} + 152923942 p^{2} T^{11} + 15841175 p^{3} T^{12} + 1284803 p^{4} T^{13} + 112237 p^{5} T^{14} + 7086 p^{6} T^{15} + 498 p^{7} T^{16} + 20 p^{8} T^{17} + p^{9} T^{18} \)
79 \( 1 - 8 T + 471 T^{2} - 3237 T^{3} + 106842 T^{4} - 664023 T^{5} + 15704132 T^{6} - 88577445 T^{7} + 1658985638 T^{8} - 8277689232 T^{9} + 1658985638 p T^{10} - 88577445 p^{2} T^{11} + 15704132 p^{3} T^{12} - 664023 p^{4} T^{13} + 106842 p^{5} T^{14} - 3237 p^{6} T^{15} + 471 p^{7} T^{16} - 8 p^{8} T^{17} + p^{9} T^{18} \)
83 \( 1 - T + 488 T^{2} - 1161 T^{3} + 113615 T^{4} - 399973 T^{5} + 16903319 T^{6} - 70850779 T^{7} + 1823276283 T^{8} - 7479201516 T^{9} + 1823276283 p T^{10} - 70850779 p^{2} T^{11} + 16903319 p^{3} T^{12} - 399973 p^{4} T^{13} + 113615 p^{5} T^{14} - 1161 p^{6} T^{15} + 488 p^{7} T^{16} - p^{8} T^{17} + p^{9} T^{18} \)
89 \( 1 - 36 T + 1226 T^{2} - 26142 T^{3} + 5835 p T^{4} - 7977763 T^{5} + 115284093 T^{6} - 1381129790 T^{7} + 15698593261 T^{8} - 151798632666 T^{9} + 15698593261 p T^{10} - 1381129790 p^{2} T^{11} + 115284093 p^{3} T^{12} - 7977763 p^{4} T^{13} + 5835 p^{6} T^{14} - 26142 p^{6} T^{15} + 1226 p^{7} T^{16} - 36 p^{8} T^{17} + p^{9} T^{18} \)
97 \( 1 - 31 T + 713 T^{2} - 13605 T^{3} + 225439 T^{4} - 3322243 T^{5} + 44076123 T^{6} - 534128975 T^{7} + 5960297404 T^{8} - 60929995076 T^{9} + 5960297404 p T^{10} - 534128975 p^{2} T^{11} + 44076123 p^{3} T^{12} - 3322243 p^{4} T^{13} + 225439 p^{5} T^{14} - 13605 p^{6} T^{15} + 713 p^{7} T^{16} - 31 p^{8} T^{17} + p^{9} T^{18} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{18} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−2.70607886251700332757295447270, −2.66671891052833934729302760557, −2.58610033244942366837803721436, −2.53454766841509965318641306742, −2.46324807194289657055643983702, −2.38089769095761072093295443866, −2.27901904670486966073280333389, −1.95857005234615926869685949532, −1.86698230614978808121087263497, −1.81704421664154926594113129424, −1.77357158003425738843707968826, −1.77307071402811282840477986323, −1.63502182877443115419705333111, −1.57868081602622836298929187307, −1.53443944429900098128924125306, −1.46059148731946406086101808548, −1.21267317792877513237501782715, −0.892792182216449293807473103113, −0.71984961285225384124660914258, −0.67674519476775632653716304034, −0.64893896729074391839953749161, −0.60014512933024843887347739797, −0.50421588409354719836052256645, −0.45405927250343831920566649579, −0.05609283255498602513243611204, 0.05609283255498602513243611204, 0.45405927250343831920566649579, 0.50421588409354719836052256645, 0.60014512933024843887347739797, 0.64893896729074391839953749161, 0.67674519476775632653716304034, 0.71984961285225384124660914258, 0.892792182216449293807473103113, 1.21267317792877513237501782715, 1.46059148731946406086101808548, 1.53443944429900098128924125306, 1.57868081602622836298929187307, 1.63502182877443115419705333111, 1.77307071402811282840477986323, 1.77357158003425738843707968826, 1.81704421664154926594113129424, 1.86698230614978808121087263497, 1.95857005234615926869685949532, 2.27901904670486966073280333389, 2.38089769095761072093295443866, 2.46324807194289657055643983702, 2.53454766841509965318641306742, 2.58610033244942366837803721436, 2.66671891052833934729302760557, 2.70607886251700332757295447270

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

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