Dirichlet series
| L(s) = 1 | − 3·2-s + 9·3-s − 2·4-s + 9·5-s − 27·6-s − 5·7-s + 11·8-s + 45·9-s − 27·10-s − 18·12-s − 9·13-s + 15·14-s + 81·15-s + 7·16-s − 5·17-s − 135·18-s − 12·19-s − 18·20-s − 45·21-s − 21·23-s + 99·24-s + 45·25-s + 27·26-s + 165·27-s + 10·28-s − 15·29-s − 243·30-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 5.19·3-s − 4-s + 4.02·5-s − 11.0·6-s − 1.88·7-s + 3.88·8-s + 15·9-s − 8.53·10-s − 5.19·12-s − 2.49·13-s + 4.00·14-s + 20.9·15-s + 7/4·16-s − 1.21·17-s − 31.8·18-s − 2.75·19-s − 4.02·20-s − 9.81·21-s − 4.37·23-s + 20.2·24-s + 9·25-s + 5.29·26-s + 31.7·27-s + 1.88·28-s − 2.78·29-s − 44.3·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{9} \cdot 5^{9} \cdot 13^{9} \cdot 31^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{9} \cdot 5^{9} \cdot 13^{9} \cdot 31^{9}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(18\) |
| Conductor: | \(3^{9} \cdot 5^{9} \cdot 13^{9} \cdot 31^{9}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(1.42252\times 10^{15}\) |
| Root analytic conductor: | \(6.94763\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(9\) |
| Selberg data: | \((18,\ 3^{9} \cdot 5^{9} \cdot 13^{9} \cdot 31^{9} ,\ ( \ : [1/2]^{9} ),\ -1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 - T )^{9} \) |
| 5 | \( ( 1 - T )^{9} \) | |
| 13 | \( ( 1 + T )^{9} \) | |
| 31 | \( ( 1 + T )^{9} \) | |
| good | 2 | \( 1 + 3 T + 11 T^{2} + 7 p^{2} T^{3} + 33 p T^{4} + 67 p T^{5} + 257 T^{6} + 435 T^{7} + 175 p^{2} T^{8} + 1021 T^{9} + 175 p^{3} T^{10} + 435 p^{2} T^{11} + 257 p^{3} T^{12} + 67 p^{5} T^{13} + 33 p^{6} T^{14} + 7 p^{8} T^{15} + 11 p^{7} T^{16} + 3 p^{8} T^{17} + p^{9} T^{18} \) |
| 7 | \( 1 + 5 T + 57 T^{2} + 237 T^{3} + 1472 T^{4} + 738 p T^{5} + 22744 T^{6} + 67486 T^{7} + 33169 p T^{8} + 577104 T^{9} + 33169 p^{2} T^{10} + 67486 p^{2} T^{11} + 22744 p^{3} T^{12} + 738 p^{5} T^{13} + 1472 p^{5} T^{14} + 237 p^{6} T^{15} + 57 p^{7} T^{16} + 5 p^{8} T^{17} + p^{9} T^{18} \) | |
| 11 | \( 1 + 4 p T^{2} - 70 T^{3} + 1003 T^{4} - 2648 T^{5} + 17661 T^{6} - 49734 T^{7} + 251882 T^{8} - 635156 T^{9} + 251882 p T^{10} - 49734 p^{2} T^{11} + 17661 p^{3} T^{12} - 2648 p^{4} T^{13} + 1003 p^{5} T^{14} - 70 p^{6} T^{15} + 4 p^{8} T^{16} + p^{9} T^{18} \) | |
| 17 | \( 1 + 5 T + 96 T^{2} + 429 T^{3} + 4610 T^{4} + 18537 T^{5} + 145338 T^{6} + 524605 T^{7} + 3312252 T^{8} + 10500230 T^{9} + 3312252 p T^{10} + 524605 p^{2} T^{11} + 145338 p^{3} T^{12} + 18537 p^{4} T^{13} + 4610 p^{5} T^{14} + 429 p^{6} T^{15} + 96 p^{7} T^{16} + 5 p^{8} T^{17} + p^{9} T^{18} \) | |
| 19 | \( 1 + 12 T + 99 T^{2} + 30 p T^{3} + 3011 T^{4} + 14274 T^{5} + 74779 T^{6} + 380688 T^{7} + 105011 p T^{8} + 9087580 T^{9} + 105011 p^{2} T^{10} + 380688 p^{2} T^{11} + 74779 p^{3} T^{12} + 14274 p^{4} T^{13} + 3011 p^{5} T^{14} + 30 p^{7} T^{15} + 99 p^{7} T^{16} + 12 p^{8} T^{17} + p^{9} T^{18} \) | |
| 23 | \( 1 + 21 T + 352 T^{2} + 4030 T^{3} + 39880 T^{4} + 321562 T^{5} + 2322771 T^{6} + 14473143 T^{7} + 82250359 T^{8} + 411866056 T^{9} + 82250359 p T^{10} + 14473143 p^{2} T^{11} + 2322771 p^{3} T^{12} + 321562 p^{4} T^{13} + 39880 p^{5} T^{14} + 4030 p^{6} T^{15} + 352 p^{7} T^{16} + 21 p^{8} T^{17} + p^{9} T^{18} \) | |
| 29 | \( 1 + 15 T + 172 T^{2} + 1279 T^{3} + 298 p T^{4} + 51148 T^{5} + 320390 T^{6} + 2166 p^{2} T^{7} + 10638354 T^{8} + 55321138 T^{9} + 10638354 p T^{10} + 2166 p^{4} T^{11} + 320390 p^{3} T^{12} + 51148 p^{4} T^{13} + 298 p^{6} T^{14} + 1279 p^{6} T^{15} + 172 p^{7} T^{16} + 15 p^{8} T^{17} + p^{9} T^{18} \) | |
| 37 | \( 1 + 13 T + 230 T^{2} + 2369 T^{3} + 25949 T^{4} + 217240 T^{5} + 49604 p T^{6} + 13055096 T^{7} + 91633307 T^{8} + 562245230 T^{9} + 91633307 p T^{10} + 13055096 p^{2} T^{11} + 49604 p^{4} T^{12} + 217240 p^{4} T^{13} + 25949 p^{5} T^{14} + 2369 p^{6} T^{15} + 230 p^{7} T^{16} + 13 p^{8} T^{17} + p^{9} T^{18} \) | |
| 41 | \( 1 + 11 T + 69 T^{2} + 258 T^{3} - 586 T^{4} - 15317 T^{5} - 39519 T^{6} + 72870 T^{7} + 4629282 T^{8} + 46527646 T^{9} + 4629282 p T^{10} + 72870 p^{2} T^{11} - 39519 p^{3} T^{12} - 15317 p^{4} T^{13} - 586 p^{5} T^{14} + 258 p^{6} T^{15} + 69 p^{7} T^{16} + 11 p^{8} T^{17} + p^{9} T^{18} \) | |
| 43 | \( 1 + 26 T + 494 T^{2} + 6604 T^{3} + 76653 T^{4} + 748365 T^{5} + 6712464 T^{6} + 53471378 T^{7} + 399148131 T^{8} + 2695839498 T^{9} + 399148131 p T^{10} + 53471378 p^{2} T^{11} + 6712464 p^{3} T^{12} + 748365 p^{4} T^{13} + 76653 p^{5} T^{14} + 6604 p^{6} T^{15} + 494 p^{7} T^{16} + 26 p^{8} T^{17} + p^{9} T^{18} \) | |
| 47 | \( 1 + 21 T + 553 T^{2} + 7851 T^{3} + 119484 T^{4} + 1281203 T^{5} + 14106542 T^{6} + 119696619 T^{7} + 1027292313 T^{8} + 7016410372 T^{9} + 1027292313 p T^{10} + 119696619 p^{2} T^{11} + 14106542 p^{3} T^{12} + 1281203 p^{4} T^{13} + 119484 p^{5} T^{14} + 7851 p^{6} T^{15} + 553 p^{7} T^{16} + 21 p^{8} T^{17} + p^{9} T^{18} \) | |
| 53 | \( 1 - 18 T + 346 T^{2} - 3587 T^{3} + 39541 T^{4} - 300760 T^{5} + 2777314 T^{6} - 19902155 T^{7} + 179692251 T^{8} - 1204842118 T^{9} + 179692251 p T^{10} - 19902155 p^{2} T^{11} + 2777314 p^{3} T^{12} - 300760 p^{4} T^{13} + 39541 p^{5} T^{14} - 3587 p^{6} T^{15} + 346 p^{7} T^{16} - 18 p^{8} T^{17} + p^{9} T^{18} \) | |
| 59 | \( 1 + 24 T + 608 T^{2} + 9483 T^{3} + 143623 T^{4} + 1690806 T^{5} + 19093649 T^{6} + 180916580 T^{7} + 1644078538 T^{8} + 12890317302 T^{9} + 1644078538 p T^{10} + 180916580 p^{2} T^{11} + 19093649 p^{3} T^{12} + 1690806 p^{4} T^{13} + 143623 p^{5} T^{14} + 9483 p^{6} T^{15} + 608 p^{7} T^{16} + 24 p^{8} T^{17} + p^{9} T^{18} \) | |
| 61 | \( 1 + 4 T + 215 T^{2} + 1414 T^{3} + 27958 T^{4} + 233890 T^{5} + 2533608 T^{6} + 24913092 T^{7} + 184343481 T^{8} + 1802227950 T^{9} + 184343481 p T^{10} + 24913092 p^{2} T^{11} + 2533608 p^{3} T^{12} + 233890 p^{4} T^{13} + 27958 p^{5} T^{14} + 1414 p^{6} T^{15} + 215 p^{7} T^{16} + 4 p^{8} T^{17} + p^{9} T^{18} \) | |
| 67 | \( 1 + 20 T + 322 T^{2} + 3931 T^{3} + 40781 T^{4} + 393806 T^{5} + 3501928 T^{6} + 29623749 T^{7} + 246047197 T^{8} + 1964890708 T^{9} + 246047197 p T^{10} + 29623749 p^{2} T^{11} + 3501928 p^{3} T^{12} + 393806 p^{4} T^{13} + 40781 p^{5} T^{14} + 3931 p^{6} T^{15} + 322 p^{7} T^{16} + 20 p^{8} T^{17} + p^{9} T^{18} \) | |
| 71 | \( 1 + 13 T + 6 p T^{2} + 4681 T^{3} + 88394 T^{4} + 846234 T^{5} + 11948845 T^{6} + 99424923 T^{7} + 1152524519 T^{8} + 8282865646 T^{9} + 1152524519 p T^{10} + 99424923 p^{2} T^{11} + 11948845 p^{3} T^{12} + 846234 p^{4} T^{13} + 88394 p^{5} T^{14} + 4681 p^{6} T^{15} + 6 p^{8} T^{16} + 13 p^{8} T^{17} + p^{9} T^{18} \) | |
| 73 | \( 1 - 15 T + 409 T^{2} - 3834 T^{3} + 59954 T^{4} - 318375 T^{5} + 3850208 T^{6} + 54673 T^{7} + 91328903 T^{8} + 17557468 p T^{9} + 91328903 p T^{10} + 54673 p^{2} T^{11} + 3850208 p^{3} T^{12} - 318375 p^{4} T^{13} + 59954 p^{5} T^{14} - 3834 p^{6} T^{15} + 409 p^{7} T^{16} - 15 p^{8} T^{17} + p^{9} T^{18} \) | |
| 79 | \( 1 + 15 T + 445 T^{2} + 5783 T^{3} + 93019 T^{4} + 1021063 T^{5} + 12321962 T^{6} + 114544383 T^{7} + 1205592346 T^{8} + 9887775904 T^{9} + 1205592346 p T^{10} + 114544383 p^{2} T^{11} + 12321962 p^{3} T^{12} + 1021063 p^{4} T^{13} + 93019 p^{5} T^{14} + 5783 p^{6} T^{15} + 445 p^{7} T^{16} + 15 p^{8} T^{17} + p^{9} T^{18} \) | |
| 83 | \( 1 + 5 T + 4 p T^{2} + 1845 T^{3} + 61639 T^{4} + 349470 T^{5} + 8329386 T^{6} + 44405270 T^{7} + 869454263 T^{8} + 4179329088 T^{9} + 869454263 p T^{10} + 44405270 p^{2} T^{11} + 8329386 p^{3} T^{12} + 349470 p^{4} T^{13} + 61639 p^{5} T^{14} + 1845 p^{6} T^{15} + 4 p^{8} T^{16} + 5 p^{8} T^{17} + p^{9} T^{18} \) | |
| 89 | \( 1 - 31 T + 545 T^{2} - 6249 T^{3} + 52948 T^{4} - 292404 T^{5} + 990786 T^{6} - 2183976 T^{7} + 73780943 T^{8} - 970405202 T^{9} + 73780943 p T^{10} - 2183976 p^{2} T^{11} + 990786 p^{3} T^{12} - 292404 p^{4} T^{13} + 52948 p^{5} T^{14} - 6249 p^{6} T^{15} + 545 p^{7} T^{16} - 31 p^{8} T^{17} + p^{9} T^{18} \) | |
| 97 | \( 1 + 11 T + 312 T^{2} + 2960 T^{3} + 46950 T^{4} + 246022 T^{5} + 2985909 T^{6} - 8069923 T^{7} - 5718697 T^{8} - 2885202234 T^{9} - 5718697 p T^{10} - 8069923 p^{2} T^{11} + 2985909 p^{3} T^{12} + 246022 p^{4} T^{13} + 46950 p^{5} T^{14} + 2960 p^{6} T^{15} + 312 p^{7} T^{16} + 11 p^{8} T^{17} + p^{9} T^{18} \) | |
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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
−3.35096840218116708819367953014, −3.20045335120152770057655355709, −3.17546683257528588941337915030, −3.13761747561052334330297729408, −3.00696856186633287969607213016, −2.95663049670082320578789285126, −2.72774823767402158882450681435, −2.67438532152874386559086125969, −2.55117742441826165480712548138, −2.48652890230990022115907473217, −2.40017831454501391042699709329, −2.28565128005416089236055237812, −2.20168748038139371251421815378, −2.11359119133213298749980712493, −2.10643142738580510394404111290, −1.98059395430779290012101433868, −1.78433242088367736110658346253, −1.70725255655970000036671300542, −1.70317715583948610159362058536, −1.45764827568081959631690491189, −1.31179856688213196032033022459, −1.30613194375693015971823149844, −1.30573694713550115863191049646, −1.26362942153030701964321795314, −1.20554042333624341558112879685, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.20554042333624341558112879685, 1.26362942153030701964321795314, 1.30573694713550115863191049646, 1.30613194375693015971823149844, 1.31179856688213196032033022459, 1.45764827568081959631690491189, 1.70317715583948610159362058536, 1.70725255655970000036671300542, 1.78433242088367736110658346253, 1.98059395430779290012101433868, 2.10643142738580510394404111290, 2.11359119133213298749980712493, 2.20168748038139371251421815378, 2.28565128005416089236055237812, 2.40017831454501391042699709329, 2.48652890230990022115907473217, 2.55117742441826165480712548138, 2.67438532152874386559086125969, 2.72774823767402158882450681435, 2.95663049670082320578789285126, 3.00696856186633287969607213016, 3.13761747561052334330297729408, 3.17546683257528588941337915030, 3.20045335120152770057655355709, 3.35096840218116708819367953014
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.