Dirichlet series
| L(s) = 1 | + 12·2-s + 18·3-s + 84·4-s + 20·5-s + 216·6-s + 26·7-s + 448·8-s + 189·9-s + 240·10-s + 63·11-s + 1.51e3·12-s + 93·13-s + 312·14-s + 360·15-s + 2.01e3·16-s + 230·17-s + 2.26e3·18-s + 89·19-s + 1.68e3·20-s + 468·21-s + 756·22-s + 81·23-s + 8.06e3·24-s − 23·25-s + 1.11e3·26-s + 1.51e3·27-s + 2.18e3·28-s + ⋯ |
| L(s) = 1 | + 4.24·2-s + 3.46·3-s + 21/2·4-s + 1.78·5-s + 14.6·6-s + 1.40·7-s + 19.7·8-s + 7·9-s + 7.58·10-s + 1.72·11-s + 36.3·12-s + 1.98·13-s + 5.95·14-s + 6.19·15-s + 63/2·16-s + 3.28·17-s + 29.6·18-s + 1.07·19-s + 18.7·20-s + 4.86·21-s + 7.32·22-s + 0.734·23-s + 68.5·24-s − 0.183·25-s + 8.41·26-s + 10.7·27-s + 14.7·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 59^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 59^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(2^{6} \cdot 3^{6} \cdot 59^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(8.30263\times 10^{7}\) |
| Root analytic conductor: | \(4.57019\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 2^{6} \cdot 3^{6} \cdot 59^{6} ,\ ( \ : [3/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(4379.714307\) |
| \(L(\frac12)\) | \(\approx\) | \(4379.714307\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - p T )^{6} \) |
| 3 | \( ( 1 - p T )^{6} \) | |
| 59 | \( ( 1 + p T )^{6} \) | |
| good | 5 | \( 1 - 4 p T + 423 T^{2} - 788 p T^{3} + 28933 T^{4} + 38548 T^{5} - 1656178 T^{6} + 38548 p^{3} T^{7} + 28933 p^{6} T^{8} - 788 p^{10} T^{9} + 423 p^{12} T^{10} - 4 p^{16} T^{11} + p^{18} T^{12} \) |
| 7 | \( 1 - 26 T + 216 p T^{2} - 30223 T^{3} + 938727 T^{4} - 15799481 T^{5} + 369349184 T^{6} - 15799481 p^{3} T^{7} + 938727 p^{6} T^{8} - 30223 p^{9} T^{9} + 216 p^{13} T^{10} - 26 p^{15} T^{11} + p^{18} T^{12} \) | |
| 11 | \( 1 - 63 T + 6257 T^{2} - 363706 T^{3} + 19078471 T^{4} - 881541851 T^{5} + 33353959390 T^{6} - 881541851 p^{3} T^{7} + 19078471 p^{6} T^{8} - 363706 p^{9} T^{9} + 6257 p^{12} T^{10} - 63 p^{15} T^{11} + p^{18} T^{12} \) | |
| 13 | \( 1 - 93 T + 7847 T^{2} - 430806 T^{3} + 1447853 p T^{4} - 610024863 T^{5} + 28449584810 T^{6} - 610024863 p^{3} T^{7} + 1447853 p^{7} T^{8} - 430806 p^{9} T^{9} + 7847 p^{12} T^{10} - 93 p^{15} T^{11} + p^{18} T^{12} \) | |
| 17 | \( 1 - 230 T + 29124 T^{2} - 2729509 T^{3} + 227491381 T^{4} - 18601726877 T^{5} + 1399773139196 T^{6} - 18601726877 p^{3} T^{7} + 227491381 p^{6} T^{8} - 2729509 p^{9} T^{9} + 29124 p^{12} T^{10} - 230 p^{15} T^{11} + p^{18} T^{12} \) | |
| 19 | \( 1 - 89 T + 14675 T^{2} - 1087923 T^{3} + 89925703 T^{4} - 8169902638 T^{5} + 500083591690 T^{6} - 8169902638 p^{3} T^{7} + 89925703 p^{6} T^{8} - 1087923 p^{9} T^{9} + 14675 p^{12} T^{10} - 89 p^{15} T^{11} + p^{18} T^{12} \) | |
| 23 | \( 1 - 81 T + 2179 p T^{2} - 2734613 T^{3} + 1158201091 T^{4} - 46487998372 T^{5} + 16966800580462 T^{6} - 46487998372 p^{3} T^{7} + 1158201091 p^{6} T^{8} - 2734613 p^{9} T^{9} + 2179 p^{13} T^{10} - 81 p^{15} T^{11} + p^{18} T^{12} \) | |
| 29 | \( 1 - 131 T + 83277 T^{2} - 13381243 T^{3} + 3849409189 T^{4} - 589518824246 T^{5} + 113454896277722 T^{6} - 589518824246 p^{3} T^{7} + 3849409189 p^{6} T^{8} - 13381243 p^{9} T^{9} + 83277 p^{12} T^{10} - 131 p^{15} T^{11} + p^{18} T^{12} \) | |
| 31 | \( 1 - 51 T + 64395 T^{2} + 577691 T^{3} + 2493107877 T^{4} - 94406801370 T^{5} + 99037364485662 T^{6} - 94406801370 p^{3} T^{7} + 2493107877 p^{6} T^{8} + 577691 p^{9} T^{9} + 64395 p^{12} T^{10} - 51 p^{15} T^{11} + p^{18} T^{12} \) | |
| 37 | \( 1 + 16 T + 181724 T^{2} + 18991415 T^{3} + 15320820519 T^{4} + 2337370147491 T^{5} + 894026837385732 T^{6} + 2337370147491 p^{3} T^{7} + 15320820519 p^{6} T^{8} + 18991415 p^{9} T^{9} + 181724 p^{12} T^{10} + 16 p^{15} T^{11} + p^{18} T^{12} \) | |
| 41 | \( 1 - 176 T + 233574 T^{2} - 19803553 T^{3} + 25255389433 T^{4} - 1229740720031 T^{5} + 1988063251221200 T^{6} - 1229740720031 p^{3} T^{7} + 25255389433 p^{6} T^{8} - 19803553 p^{9} T^{9} + 233574 p^{12} T^{10} - 176 p^{15} T^{11} + p^{18} T^{12} \) | |
| 43 | \( 1 - 375 T + 241877 T^{2} - 49534686 T^{3} + 28305286691 T^{4} - 6268661198259 T^{5} + 69747413882666 p T^{6} - 6268661198259 p^{3} T^{7} + 28305286691 p^{6} T^{8} - 49534686 p^{9} T^{9} + 241877 p^{12} T^{10} - 375 p^{15} T^{11} + p^{18} T^{12} \) | |
| 47 | \( 1 + 255 T + 216869 T^{2} + 50471521 T^{3} + 23585281147 T^{4} + 2082359199602 T^{5} + 2211300095928766 T^{6} + 2082359199602 p^{3} T^{7} + 23585281147 p^{6} T^{8} + 50471521 p^{9} T^{9} + 216869 p^{12} T^{10} + 255 p^{15} T^{11} + p^{18} T^{12} \) | |
| 53 | \( 1 + 256 T + 524559 T^{2} + 76610212 T^{3} + 127575492837 T^{4} + 12532960412320 T^{5} + 21813976397959054 T^{6} + 12532960412320 p^{3} T^{7} + 127575492837 p^{6} T^{8} + 76610212 p^{9} T^{9} + 524559 p^{12} T^{10} + 256 p^{15} T^{11} + p^{18} T^{12} \) | |
| 61 | \( 1 - 39 T + 477461 T^{2} - 41242735 T^{3} + 104229311789 T^{4} - 10165622492558 T^{5} + 19694680388001914 T^{6} - 10165622492558 p^{3} T^{7} + 104229311789 p^{6} T^{8} - 41242735 p^{9} T^{9} + 477461 p^{12} T^{10} - 39 p^{15} T^{11} + p^{18} T^{12} \) | |
| 67 | \( 1 - 86 T + 1376543 T^{2} - 141947178 T^{3} + 882035807755 T^{4} - 84377047671916 T^{5} + 335782995070613386 T^{6} - 84377047671916 p^{3} T^{7} + 882035807755 p^{6} T^{8} - 141947178 p^{9} T^{9} + 1376543 p^{12} T^{10} - 86 p^{15} T^{11} + p^{18} T^{12} \) | |
| 71 | \( 1 + 895 T + 2169489 T^{2} + 1416156292 T^{3} + 1892304587895 T^{4} + 940898752283695 T^{5} + 890214151633467118 T^{6} + 940898752283695 p^{3} T^{7} + 1892304587895 p^{6} T^{8} + 1416156292 p^{9} T^{9} + 2169489 p^{12} T^{10} + 895 p^{15} T^{11} + p^{18} T^{12} \) | |
| 73 | \( 1 - 155 T + 803291 T^{2} - 150892557 T^{3} + 333226933471 T^{4} - 161787754591408 T^{5} + 131066806595063242 T^{6} - 161787754591408 p^{3} T^{7} + 333226933471 p^{6} T^{8} - 150892557 p^{9} T^{9} + 803291 p^{12} T^{10} - 155 p^{15} T^{11} + p^{18} T^{12} \) | |
| 79 | \( 1 + 565 T + 2095661 T^{2} + 814032494 T^{3} + 2047815974859 T^{4} + 633867119732397 T^{5} + 1257584657823203598 T^{6} + 633867119732397 p^{3} T^{7} + 2047815974859 p^{6} T^{8} + 814032494 p^{9} T^{9} + 2095661 p^{12} T^{10} + 565 p^{15} T^{11} + p^{18} T^{12} \) | |
| 83 | \( 1 - 140 T + 1337014 T^{2} - 335353869 T^{3} + 1233155625427 T^{4} - 357199616830977 T^{5} + 757734507811477692 T^{6} - 357199616830977 p^{3} T^{7} + 1233155625427 p^{6} T^{8} - 335353869 p^{9} T^{9} + 1337014 p^{12} T^{10} - 140 p^{15} T^{11} + p^{18} T^{12} \) | |
| 89 | \( 1 + 1769 T + 1890363 T^{2} + 1247784455 T^{3} + 1012593301251 T^{4} + 992564405238992 T^{5} + 1069254273205666834 T^{6} + 992564405238992 p^{3} T^{7} + 1012593301251 p^{6} T^{8} + 1247784455 p^{9} T^{9} + 1890363 p^{12} T^{10} + 1769 p^{15} T^{11} + p^{18} T^{12} \) | |
| 97 | \( 1 - 48 T + 3268959 T^{2} - 75889774 T^{3} + 5522378992239 T^{4} - 100481357758866 T^{5} + 6129478607568321474 T^{6} - 100481357758866 p^{3} T^{7} + 5522378992239 p^{6} T^{8} - 75889774 p^{9} T^{9} + 3268959 p^{12} T^{10} - 48 p^{15} T^{11} + p^{18} T^{12} \) | |
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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
−5.87205273462534289048416723260, −5.35885922677527094054418214195, −5.24041597094398866342591030610, −5.00139993764991287962118815910, −4.97364944432932615478524717398, −4.92403808670813748870324234190, −4.72109508247669127291476929124, −4.07025969162151525484305744149, −3.98249948594592295646067540254, −3.89400670282754261525938130061, −3.86975078166070024109616020851, −3.85431268799719572372869566658, −3.52223561796638240572051076763, −3.10780439703371689804573697401, −3.05437799873637390157953675267, −2.89935780055347289040488970392, −2.60096096387622096766341504944, −2.56102475938474141620696815794, −2.44695443378289547167744716768, −1.69941455672594034825578424932, −1.53754866880028251268687920957, −1.53714635485635091826750530674, −1.33536586434539823265113553730, −1.31203364921896348379345344209, −1.13411883606143262894636093403, 1.13411883606143262894636093403, 1.31203364921896348379345344209, 1.33536586434539823265113553730, 1.53714635485635091826750530674, 1.53754866880028251268687920957, 1.69941455672594034825578424932, 2.44695443378289547167744716768, 2.56102475938474141620696815794, 2.60096096387622096766341504944, 2.89935780055347289040488970392, 3.05437799873637390157953675267, 3.10780439703371689804573697401, 3.52223561796638240572051076763, 3.85431268799719572372869566658, 3.86975078166070024109616020851, 3.89400670282754261525938130061, 3.98249948594592295646067540254, 4.07025969162151525484305744149, 4.72109508247669127291476929124, 4.92403808670813748870324234190, 4.97364944432932615478524717398, 5.00139993764991287962118815910, 5.24041597094398866342591030610, 5.35885922677527094054418214195, 5.87205273462534289048416723260
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.