Dirichlet series
| L(s) = 1 | + 2·3-s − 43·7-s − 47·9-s − 15·11-s − 87·13-s − 76·17-s − 21·19-s − 86·21-s − 91·23-s − 140·27-s − 86·29-s + 246·31-s − 30·33-s − 1.05e3·37-s − 174·39-s − 469·41-s − 460·43-s − 1.32e3·47-s + 122·49-s − 152·51-s − 1.49e3·53-s − 42·57-s + 763·59-s + 522·61-s + 2.02e3·63-s − 1.09e3·67-s − 182·69-s + ⋯ |
| L(s) = 1 | + 0.384·3-s − 2.32·7-s − 1.74·9-s − 0.411·11-s − 1.85·13-s − 1.08·17-s − 0.253·19-s − 0.893·21-s − 0.824·23-s − 0.997·27-s − 0.550·29-s + 1.42·31-s − 0.158·33-s − 4.67·37-s − 0.714·39-s − 1.78·41-s − 1.63·43-s − 4.09·47-s + 0.355·49-s − 0.417·51-s − 3.88·53-s − 0.0975·57-s + 1.68·59-s + 1.09·61-s + 4.04·63-s − 2.00·67-s − 0.317·69-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{18}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(2^{12} \cdot 5^{18}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.59198\times 10^{8}\) |
| Root analytic conductor: | \(5.43147\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(6\) |
| Selberg data: | \((12,\ 2^{12} \cdot 5^{18} ,\ ( \ : [3/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) | |
| good | 3 | \( 1 - 2 T + 17 p T^{2} - 56 T^{3} + 512 p T^{4} - 5468 T^{5} + 53708 T^{6} - 5468 p^{3} T^{7} + 512 p^{7} T^{8} - 56 p^{9} T^{9} + 17 p^{13} T^{10} - 2 p^{15} T^{11} + p^{18} T^{12} \) |
| 7 | \( 1 + 43 T + 1727 T^{2} + 50339 T^{3} + 27198 p^{2} T^{4} + 29618219 T^{5} + 588366411 T^{6} + 29618219 p^{3} T^{7} + 27198 p^{8} T^{8} + 50339 p^{9} T^{9} + 1727 p^{12} T^{10} + 43 p^{15} T^{11} + p^{18} T^{12} \) | |
| 11 | \( 1 + 15 T + 2571 T^{2} - 925 T^{3} + 4783030 T^{4} + 26932775 T^{5} + 9069914455 T^{6} + 26932775 p^{3} T^{7} + 4783030 p^{6} T^{8} - 925 p^{9} T^{9} + 2571 p^{12} T^{10} + 15 p^{15} T^{11} + p^{18} T^{12} \) | |
| 13 | \( 1 + 87 T + 14276 T^{2} + 67902 p T^{3} + 79599801 T^{4} + 3692238618 T^{5} + 233059547893 T^{6} + 3692238618 p^{3} T^{7} + 79599801 p^{6} T^{8} + 67902 p^{10} T^{9} + 14276 p^{12} T^{10} + 87 p^{15} T^{11} + p^{18} T^{12} \) | |
| 17 | \( 1 + 76 T + 24044 T^{2} + 1635932 T^{3} + 265844896 T^{4} + 15066200524 T^{5} + 1683395320122 T^{6} + 15066200524 p^{3} T^{7} + 265844896 p^{6} T^{8} + 1635932 p^{9} T^{9} + 24044 p^{12} T^{10} + 76 p^{15} T^{11} + p^{18} T^{12} \) | |
| 19 | \( 1 + 21 T + 18384 T^{2} + 952000 T^{3} + 114342925 T^{4} + 14391271636 T^{5} + 482928454281 T^{6} + 14391271636 p^{3} T^{7} + 114342925 p^{6} T^{8} + 952000 p^{9} T^{9} + 18384 p^{12} T^{10} + 21 p^{15} T^{11} + p^{18} T^{12} \) | |
| 23 | \( 1 + 91 T + 43183 T^{2} + 4465237 T^{3} + 1043591407 T^{4} + 89603116498 T^{5} + 16186863234834 T^{6} + 89603116498 p^{3} T^{7} + 1043591407 p^{6} T^{8} + 4465237 p^{9} T^{9} + 43183 p^{12} T^{10} + 91 p^{15} T^{11} + p^{18} T^{12} \) | |
| 29 | \( 1 + 86 T + 40984 T^{2} + 1546130 T^{3} + 1139137440 T^{4} + 12068851726 T^{5} + 24261520960446 T^{6} + 12068851726 p^{3} T^{7} + 1139137440 p^{6} T^{8} + 1546130 p^{9} T^{9} + 40984 p^{12} T^{10} + 86 p^{15} T^{11} + p^{18} T^{12} \) | |
| 31 | \( 1 - 246 T + 102211 T^{2} - 21255600 T^{3} + 4958800640 T^{4} - 1053332733876 T^{5} + 177504262654556 T^{6} - 1053332733876 p^{3} T^{7} + 4958800640 p^{6} T^{8} - 21255600 p^{9} T^{9} + 102211 p^{12} T^{10} - 246 p^{15} T^{11} + p^{18} T^{12} \) | |
| 37 | \( 1 + 1053 T + 617027 T^{2} + 268726669 T^{3} + 94628719287 T^{4} + 27439323210994 T^{5} + 6692495838196826 T^{6} + 27439323210994 p^{3} T^{7} + 94628719287 p^{6} T^{8} + 268726669 p^{9} T^{9} + 617027 p^{12} T^{10} + 1053 p^{15} T^{11} + p^{18} T^{12} \) | |
| 41 | \( 1 + 469 T + 370651 T^{2} + 130519935 T^{3} + 59679335290 T^{4} + 16193636465489 T^{5} + 130666326947491 p T^{6} + 16193636465489 p^{3} T^{7} + 59679335290 p^{6} T^{8} + 130519935 p^{9} T^{9} + 370651 p^{12} T^{10} + 469 p^{15} T^{11} + p^{18} T^{12} \) | |
| 43 | \( 1 + 460 T + 345832 T^{2} + 124655500 T^{3} + 57214872680 T^{4} + 16210134848300 T^{5} + 5719975930729670 T^{6} + 16210134848300 p^{3} T^{7} + 57214872680 p^{6} T^{8} + 124655500 p^{9} T^{9} + 345832 p^{12} T^{10} + 460 p^{15} T^{11} + p^{18} T^{12} \) | |
| 47 | \( 1 + 1321 T + 1027284 T^{2} + 568530472 T^{3} + 254241148401 T^{4} + 96261836646084 T^{5} + 32639413993908497 T^{6} + 96261836646084 p^{3} T^{7} + 254241148401 p^{6} T^{8} + 568530472 p^{9} T^{9} + 1027284 p^{12} T^{10} + 1321 p^{15} T^{11} + p^{18} T^{12} \) | |
| 53 | \( 1 + 1499 T + 1482588 T^{2} + 962718538 T^{3} + 509576746097 T^{4} + 218580703559942 T^{5} + 88672564803858969 T^{6} + 218580703559942 p^{3} T^{7} + 509576746097 p^{6} T^{8} + 962718538 p^{9} T^{9} + 1482588 p^{12} T^{10} + 1499 p^{15} T^{11} + p^{18} T^{12} \) | |
| 59 | \( 1 - 763 T + 1139444 T^{2} - 637509320 T^{3} + 549686280065 T^{4} - 238249689357988 T^{5} + 147613247387932589 T^{6} - 238249689357988 p^{3} T^{7} + 549686280065 p^{6} T^{8} - 637509320 p^{9} T^{9} + 1139444 p^{12} T^{10} - 763 p^{15} T^{11} + p^{18} T^{12} \) | |
| 61 | \( 1 - 522 T + 911561 T^{2} - 447929140 T^{3} + 432230538640 T^{4} - 180312066611682 T^{5} + 122189438133494244 T^{6} - 180312066611682 p^{3} T^{7} + 432230538640 p^{6} T^{8} - 447929140 p^{9} T^{9} + 911561 p^{12} T^{10} - 522 p^{15} T^{11} + p^{18} T^{12} \) | |
| 67 | \( 1 + 1098 T + 1304607 T^{2} + 804289824 T^{3} + 574654791952 T^{4} + 280805402819484 T^{5} + 179454546747099556 T^{6} + 280805402819484 p^{3} T^{7} + 574654791952 p^{6} T^{8} + 804289824 p^{9} T^{9} + 1304607 p^{12} T^{10} + 1098 p^{15} T^{11} + p^{18} T^{12} \) | |
| 71 | \( 1 - 478 T + 1597916 T^{2} - 467705510 T^{3} + 1082902123840 T^{4} - 201429750683518 T^{5} + 458346087979780794 T^{6} - 201429750683518 p^{3} T^{7} + 1082902123840 p^{6} T^{8} - 467705510 p^{9} T^{9} + 1597916 p^{12} T^{10} - 478 p^{15} T^{11} + p^{18} T^{12} \) | |
| 73 | \( 1 + 2352 T + 3503681 T^{2} + 3738530336 T^{3} + 3327345801196 T^{4} + 2522869246421238 T^{5} + 1686917326084339788 T^{6} + 2522869246421238 p^{3} T^{7} + 3327345801196 p^{6} T^{8} + 3738530336 p^{9} T^{9} + 3503681 p^{12} T^{10} + 2352 p^{15} T^{11} + p^{18} T^{12} \) | |
| 79 | \( 1 - 1072 T + 2649204 T^{2} - 2097802080 T^{3} + 2905185093440 T^{4} - 22697510207728 p T^{5} + 1816478252253917634 T^{6} - 22697510207728 p^{4} T^{7} + 2905185093440 p^{6} T^{8} - 2097802080 p^{9} T^{9} + 2649204 p^{12} T^{10} - 1072 p^{15} T^{11} + p^{18} T^{12} \) | |
| 83 | \( 1 + 576 T + 1544378 T^{2} + 347760672 T^{3} + 596656676487 T^{4} - 276466664748192 T^{5} + 44146699494821164 T^{6} - 276466664748192 p^{3} T^{7} + 596656676487 p^{6} T^{8} + 347760672 p^{9} T^{9} + 1544378 p^{12} T^{10} + 576 p^{15} T^{11} + p^{18} T^{12} \) | |
| 89 | \( 1 + 459 T + 2736869 T^{2} + 1264607975 T^{3} + 3728008077875 T^{4} + 1598776797879014 T^{5} + 3233583446540907726 T^{6} + 1598776797879014 p^{3} T^{7} + 3728008077875 p^{6} T^{8} + 1264607975 p^{9} T^{9} + 2736869 p^{12} T^{10} + 459 p^{15} T^{11} + p^{18} T^{12} \) | |
| 97 | \( 1 + 3228 T + 7619597 T^{2} + 12374348724 T^{3} + 17384034216012 T^{4} + 19837163355251154 T^{5} + 20598257554511957836 T^{6} + 19837163355251154 p^{3} T^{7} + 17384034216012 p^{6} T^{8} + 12374348724 p^{9} T^{9} + 7619597 p^{12} T^{10} + 3228 p^{15} T^{11} + p^{18} T^{12} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.26474209816549946596923136837, −5.61682997445979530981115434932, −5.47482865127790612156454204273, −5.32404303613567204149334165450, −5.23821631948601049800127270235, −5.20482278859355307874460194629, −5.13407572612089152955921451016, −4.74866499983476307921937199818, −4.43160309592491624388278077545, −4.32166736266445889813974861070, −4.31521947325170501893178309520, −3.77332198469220408479186218817, −3.73523524048963529218152155066, −3.39498851186377264024300863099, −3.20940108735422782722254376817, −3.11027344964804098561443583213, −2.97840804324018437739451964151, −2.95079454806400052927805847600, −2.77038294196425417326529281475, −2.26572200184020302382351117699, −2.09041951695974022561667379980, −1.81780295220481254504371355637, −1.74489677981991337432367254808, −1.29675420514435912158551071481, −1.29391035826847732565052622956, 0, 0, 0, 0, 0, 0, 1.29391035826847732565052622956, 1.29675420514435912158551071481, 1.74489677981991337432367254808, 1.81780295220481254504371355637, 2.09041951695974022561667379980, 2.26572200184020302382351117699, 2.77038294196425417326529281475, 2.95079454806400052927805847600, 2.97840804324018437739451964151, 3.11027344964804098561443583213, 3.20940108735422782722254376817, 3.39498851186377264024300863099, 3.73523524048963529218152155066, 3.77332198469220408479186218817, 4.31521947325170501893178309520, 4.32166736266445889813974861070, 4.43160309592491624388278077545, 4.74866499983476307921937199818, 5.13407572612089152955921451016, 5.20482278859355307874460194629, 5.23821631948601049800127270235, 5.32404303613567204149334165450, 5.47482865127790612156454204273, 5.61682997445979530981115434932, 6.26474209816549946596923136837
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.