Dirichlet series
| L(s) = 1 | − 5·2-s + 20·3-s − 68·4-s − 150·5-s − 100·6-s + 390·8-s − 340·9-s + 750·10-s − 924·11-s − 1.36e3·12-s − 150·13-s − 3.00e3·15-s + 2.07e3·16-s + 1.54e3·17-s + 1.70e3·18-s + 92·19-s + 1.02e4·20-s + 4.62e3·22-s − 3.92e3·23-s + 7.80e3·24-s + 1.31e4·25-s + 750·26-s − 1.04e4·27-s + 1.26e3·29-s + 1.50e4·30-s − 7.16e3·31-s − 1.17e4·32-s + ⋯ |
| L(s) = 1 | − 0.883·2-s + 1.28·3-s − 2.12·4-s − 2.68·5-s − 1.13·6-s + 2.15·8-s − 1.39·9-s + 2.37·10-s − 2.30·11-s − 2.72·12-s − 0.246·13-s − 3.44·15-s + 2.03·16-s + 1.29·17-s + 1.23·18-s + 0.0584·19-s + 5.70·20-s + 2.03·22-s − 1.54·23-s + 2.76·24-s + 21/5·25-s + 0.217·26-s − 2.74·27-s + 0.279·29-s + 3.04·30-s − 1.33·31-s − 2.02·32-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(5^{6} \cdot 7^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.68094\times 10^{9}\) |
| Root analytic conductor: | \(6.26849\) |
| Motivic weight: | \(5\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(6\) |
| Selberg data: | \((12,\ 5^{6} \cdot 7^{12} ,\ ( \ : [5/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(3)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{7}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( ( 1 + p^{2} T )^{6} \) |
| 7 | \( 1 \) | |
| good | 2 | \( 1 + 5 T + 93 T^{2} + 415 T^{3} + 2185 p T^{4} + 3785 p^{2} T^{5} + 19281 p^{3} T^{6} + 3785 p^{7} T^{7} + 2185 p^{11} T^{8} + 415 p^{15} T^{9} + 93 p^{20} T^{10} + 5 p^{25} T^{11} + p^{30} T^{12} \) |
| 3 | \( 1 - 20 T + 740 T^{2} - 11200 T^{3} + 30640 p^{2} T^{4} - 482020 p^{2} T^{5} + 3044590 p^{3} T^{6} - 482020 p^{7} T^{7} + 30640 p^{12} T^{8} - 11200 p^{15} T^{9} + 740 p^{20} T^{10} - 20 p^{25} T^{11} + p^{30} T^{12} \) | |
| 11 | \( 1 + 84 p T + 825717 T^{2} + 447883644 T^{3} + 251513820115 T^{4} + 108565023355632 T^{5} + 48862291165415198 T^{6} + 108565023355632 p^{5} T^{7} + 251513820115 p^{10} T^{8} + 447883644 p^{15} T^{9} + 825717 p^{20} T^{10} + 84 p^{26} T^{11} + p^{30} T^{12} \) | |
| 13 | \( 1 + 150 T + 796287 T^{2} + 297858110 T^{3} + 464061438907 T^{4} + 133413930219380 T^{5} + 231222100617263226 T^{6} + 133413930219380 p^{5} T^{7} + 464061438907 p^{10} T^{8} + 297858110 p^{15} T^{9} + 796287 p^{20} T^{10} + 150 p^{25} T^{11} + p^{30} T^{12} \) | |
| 17 | \( 1 - 1540 T + 7451530 T^{2} - 8871686260 T^{3} + 24111152130175 T^{4} - 22932939816405640 T^{5} + 44337762427137456300 T^{6} - 22932939816405640 p^{5} T^{7} + 24111152130175 p^{10} T^{8} - 8871686260 p^{15} T^{9} + 7451530 p^{20} T^{10} - 1540 p^{25} T^{11} + p^{30} T^{12} \) | |
| 19 | \( 1 - 92 T + 7512461 T^{2} - 3685957332 T^{3} + 25688116768307 T^{4} - 23303405428560392 T^{5} + 63936283449584109166 T^{6} - 23303405428560392 p^{5} T^{7} + 25688116768307 p^{10} T^{8} - 3685957332 p^{15} T^{9} + 7512461 p^{20} T^{10} - 92 p^{25} T^{11} + p^{30} T^{12} \) | |
| 23 | \( 1 + 3920 T + 24036187 T^{2} + 81532050560 T^{3} + 328692879009670 T^{4} + 886267605267526160 T^{5} + \)\(25\!\cdots\!07\)\( T^{6} + 886267605267526160 p^{5} T^{7} + 328692879009670 p^{10} T^{8} + 81532050560 p^{15} T^{9} + 24036187 p^{20} T^{10} + 3920 p^{25} T^{11} + p^{30} T^{12} \) | |
| 29 | \( 1 - 1264 T + 34338048 T^{2} - 97759319540 T^{3} + 1265693527774624 T^{4} - 2479979403250472560 T^{5} + \)\(30\!\cdots\!54\)\( T^{6} - 2479979403250472560 p^{5} T^{7} + 1265693527774624 p^{10} T^{8} - 97759319540 p^{15} T^{9} + 34338048 p^{20} T^{10} - 1264 p^{25} T^{11} + p^{30} T^{12} \) | |
| 31 | \( 1 + 7160 T + 91720490 T^{2} + 813875493096 T^{3} + 5583909231338895 T^{4} + 36208570257510880880 T^{5} + \)\(69\!\cdots\!76\)\( p T^{6} + 36208570257510880880 p^{5} T^{7} + 5583909231338895 p^{10} T^{8} + 813875493096 p^{15} T^{9} + 91720490 p^{20} T^{10} + 7160 p^{25} T^{11} + p^{30} T^{12} \) | |
| 37 | \( 1 + 14170 T + 318813415 T^{2} + 3192181392650 T^{3} + 46176184376023275 T^{4} + \)\(36\!\cdots\!20\)\( T^{5} + \)\(40\!\cdots\!50\)\( T^{6} + \)\(36\!\cdots\!20\)\( p^{5} T^{7} + 46176184376023275 p^{10} T^{8} + 3192181392650 p^{15} T^{9} + 318813415 p^{20} T^{10} + 14170 p^{25} T^{11} + p^{30} T^{12} \) | |
| 41 | \( 1 - 4098 T + 461440565 T^{2} - 793603240022 T^{3} + 101430559965472602 T^{4} - 78439540555050663786 T^{5} + \)\(14\!\cdots\!01\)\( T^{6} - 78439540555050663786 p^{5} T^{7} + 101430559965472602 p^{10} T^{8} - 793603240022 p^{15} T^{9} + 461440565 p^{20} T^{10} - 4098 p^{25} T^{11} + p^{30} T^{12} \) | |
| 43 | \( 1 + 24460 T + 767727092 T^{2} + 12925607212560 T^{3} + 244456305714648992 T^{4} + \)\(31\!\cdots\!20\)\( T^{5} + \)\(45\!\cdots\!66\)\( T^{6} + \)\(31\!\cdots\!20\)\( p^{5} T^{7} + 244456305714648992 p^{10} T^{8} + 12925607212560 p^{15} T^{9} + 767727092 p^{20} T^{10} + 24460 p^{25} T^{11} + p^{30} T^{12} \) | |
| 47 | \( 1 - 42940 T + 1662499309 T^{2} - 39067396764460 T^{3} + 894007089501510835 T^{4} - \)\(15\!\cdots\!00\)\( T^{5} + \)\(26\!\cdots\!94\)\( T^{6} - \)\(15\!\cdots\!00\)\( p^{5} T^{7} + 894007089501510835 p^{10} T^{8} - 39067396764460 p^{15} T^{9} + 1662499309 p^{20} T^{10} - 42940 p^{25} T^{11} + p^{30} T^{12} \) | |
| 53 | \( 1 + 2450 T + 1731298775 T^{2} + 12557810325090 T^{3} + 1332901811960979875 T^{4} + \)\(13\!\cdots\!00\)\( T^{5} + \)\(65\!\cdots\!30\)\( T^{6} + \)\(13\!\cdots\!00\)\( p^{5} T^{7} + 1332901811960979875 p^{10} T^{8} + 12557810325090 p^{15} T^{9} + 1731298775 p^{20} T^{10} + 2450 p^{25} T^{11} + p^{30} T^{12} \) | |
| 59 | \( 1 + 64600 T + 4003051370 T^{2} + 159881299023080 T^{3} + 6047720347419053255 T^{4} + \)\(18\!\cdots\!00\)\( T^{5} + \)\(53\!\cdots\!48\)\( T^{6} + \)\(18\!\cdots\!00\)\( p^{5} T^{7} + 6047720347419053255 p^{10} T^{8} + 159881299023080 p^{15} T^{9} + 4003051370 p^{20} T^{10} + 64600 p^{25} T^{11} + p^{30} T^{12} \) | |
| 61 | \( 1 - 73620 T + 5595394780 T^{2} - 262435447105344 T^{3} + 11885303041254933320 T^{4} - \)\(40\!\cdots\!80\)\( T^{5} + \)\(13\!\cdots\!86\)\( T^{6} - \)\(40\!\cdots\!80\)\( p^{5} T^{7} + 11885303041254933320 p^{10} T^{8} - 262435447105344 p^{15} T^{9} + 5595394780 p^{20} T^{10} - 73620 p^{25} T^{11} + p^{30} T^{12} \) | |
| 67 | \( 1 + 142620 T + 10633871464 T^{2} + 485585751900360 T^{3} + 13732683630728961880 T^{4} + \)\(19\!\cdots\!40\)\( T^{5} + \)\(13\!\cdots\!34\)\( T^{6} + \)\(19\!\cdots\!40\)\( p^{5} T^{7} + 13732683630728961880 p^{10} T^{8} + 485585751900360 p^{15} T^{9} + 10633871464 p^{20} T^{10} + 142620 p^{25} T^{11} + p^{30} T^{12} \) | |
| 71 | \( 1 + 154256 T + 15903461602 T^{2} + 1239493386418352 T^{3} + 78922551198445875343 T^{4} + \)\(42\!\cdots\!92\)\( T^{5} + \)\(19\!\cdots\!08\)\( T^{6} + \)\(42\!\cdots\!92\)\( p^{5} T^{7} + 78922551198445875343 p^{10} T^{8} + 1239493386418352 p^{15} T^{9} + 15903461602 p^{20} T^{10} + 154256 p^{25} T^{11} + p^{30} T^{12} \) | |
| 73 | \( 1 + 5120 T + 6905477830 T^{2} + 111041937751360 T^{3} + 25601871633036081215 T^{4} + \)\(45\!\cdots\!80\)\( T^{5} + \)\(64\!\cdots\!60\)\( T^{6} + \)\(45\!\cdots\!80\)\( p^{5} T^{7} + 25601871633036081215 p^{10} T^{8} + 111041937751360 p^{15} T^{9} + 6905477830 p^{20} T^{10} + 5120 p^{25} T^{11} + p^{30} T^{12} \) | |
| 79 | \( 1 + 222504 T + 30790489134 T^{2} + 3146130332337464 T^{3} + \)\(26\!\cdots\!03\)\( T^{4} + \)\(18\!\cdots\!04\)\( T^{5} + \)\(10\!\cdots\!60\)\( T^{6} + \)\(18\!\cdots\!04\)\( p^{5} T^{7} + \)\(26\!\cdots\!03\)\( p^{10} T^{8} + 3146130332337464 p^{15} T^{9} + 30790489134 p^{20} T^{10} + 222504 p^{25} T^{11} + p^{30} T^{12} \) | |
| 83 | \( 1 - 179580 T + 26415024920 T^{2} - 2643470654451640 T^{3} + \)\(24\!\cdots\!72\)\( T^{4} - \)\(18\!\cdots\!40\)\( T^{5} + \)\(12\!\cdots\!10\)\( T^{6} - \)\(18\!\cdots\!40\)\( p^{5} T^{7} + \)\(24\!\cdots\!72\)\( p^{10} T^{8} - 2643470654451640 p^{15} T^{9} + 26415024920 p^{20} T^{10} - 179580 p^{25} T^{11} + p^{30} T^{12} \) | |
| 89 | \( 1 - 41648 T + 14985273296 T^{2} - 711975402452908 T^{3} + \)\(13\!\cdots\!52\)\( T^{4} - \)\(54\!\cdots\!48\)\( T^{5} + \)\(89\!\cdots\!46\)\( T^{6} - \)\(54\!\cdots\!48\)\( p^{5} T^{7} + \)\(13\!\cdots\!52\)\( p^{10} T^{8} - 711975402452908 p^{15} T^{9} + 14985273296 p^{20} T^{10} - 41648 p^{25} T^{11} + p^{30} T^{12} \) | |
| 97 | \( 1 + 73980 T + 24090535458 T^{2} + 2757215884896140 T^{3} + \)\(39\!\cdots\!07\)\( T^{4} + \)\(34\!\cdots\!00\)\( T^{5} + \)\(46\!\cdots\!84\)\( T^{6} + \)\(34\!\cdots\!00\)\( p^{5} T^{7} + \)\(39\!\cdots\!07\)\( p^{10} T^{8} + 2757215884896140 p^{15} T^{9} + 24090535458 p^{20} T^{10} + 73980 p^{25} T^{11} + p^{30} T^{12} \) | |
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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
−6.27477490502836555546470677380, −6.02737344276277312033647336075, −5.55496298657086659619928415561, −5.52312059150619166021419666986, −5.51552553048396383021758408746, −5.43995499433310503026776973112, −5.12510116462414021518067263054, −4.76439962533707293214923117743, −4.64021937632334924522700308637, −4.42138424010970918297167830168, −4.22747393210658824808671607627, −4.06318157292085137732634478981, −4.02608668632801916178858915846, −3.48053194199081156568177165902, −3.44864088181363275860441974354, −3.21873185990455479347471450994, −3.16387213929912410247230503341, −2.77393816521697384147327131406, −2.70189779788680767413977195466, −2.42132571445461832794180889249, −2.23991617151723724419948030948, −1.53698927422100383023646048256, −1.43751445572590652987713545016, −1.05822187658478752204260619945, −0.936751823763627328388798543442, 0, 0, 0, 0, 0, 0, 0.936751823763627328388798543442, 1.05822187658478752204260619945, 1.43751445572590652987713545016, 1.53698927422100383023646048256, 2.23991617151723724419948030948, 2.42132571445461832794180889249, 2.70189779788680767413977195466, 2.77393816521697384147327131406, 3.16387213929912410247230503341, 3.21873185990455479347471450994, 3.44864088181363275860441974354, 3.48053194199081156568177165902, 4.02608668632801916178858915846, 4.06318157292085137732634478981, 4.22747393210658824808671607627, 4.42138424010970918297167830168, 4.64021937632334924522700308637, 4.76439962533707293214923117743, 5.12510116462414021518067263054, 5.43995499433310503026776973112, 5.51552553048396383021758408746, 5.52312059150619166021419666986, 5.55496298657086659619928415561, 6.02737344276277312033647336075, 6.27477490502836555546470677380
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.