Dirichlet series
| L(s) = 1 | + 6·2-s − 7·3-s + 5·4-s − 43·5-s − 42·6-s − 40·8-s − 16·9-s − 258·10-s − 28·11-s − 35·12-s − 56·13-s + 301·15-s − 141·16-s − 19·17-s − 96·18-s + 75·19-s − 215·20-s − 168·22-s + 131·23-s + 280·24-s + 602·25-s − 336·26-s + 210·27-s + 515·29-s + 1.80e3·30-s − 237·31-s − 206·32-s + ⋯ |
| L(s) = 1 | + 2.12·2-s − 1.34·3-s + 5/8·4-s − 3.84·5-s − 2.85·6-s − 1.76·8-s − 0.592·9-s − 8.15·10-s − 0.767·11-s − 0.841·12-s − 1.19·13-s + 5.18·15-s − 2.20·16-s − 0.271·17-s − 1.25·18-s + 0.905·19-s − 2.40·20-s − 1.62·22-s + 1.18·23-s + 2.38·24-s + 4.81·25-s − 2.53·26-s + 1.49·27-s + 3.29·29-s + 10.9·30-s − 1.37·31-s − 1.13·32-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 43^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 43^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(7^{12} \cdot 43^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.69133\times 10^{12}\) |
| Root analytic conductor: | \(11.1497\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(6\) |
| Selberg data: | \((12,\ 7^{12} \cdot 43^{6} ,\ ( \ : [3/2]^{6} ),\ 1 )\) |
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 | 7 | \( 1 \) |
| 43 | \( ( 1 + p T )^{6} \) | |
| good | 2 | \( 1 - 3 p T + 31 T^{2} - 29 p^{2} T^{3} + 221 p T^{4} - 23 p^{6} T^{5} + 1167 p^{2} T^{6} - 23 p^{9} T^{7} + 221 p^{7} T^{8} - 29 p^{11} T^{9} + 31 p^{12} T^{10} - 3 p^{16} T^{11} + p^{18} T^{12} \) |
| 3 | \( 1 + 7 T + 65 T^{2} + 119 p T^{3} + 2599 T^{4} + 15158 T^{5} + 96098 T^{6} + 15158 p^{3} T^{7} + 2599 p^{6} T^{8} + 119 p^{10} T^{9} + 65 p^{12} T^{10} + 7 p^{15} T^{11} + p^{18} T^{12} \) | |
| 5 | \( 1 + 43 T + 1247 T^{2} + 26367 T^{3} + 452519 T^{4} + 6421366 T^{5} + 77975134 T^{6} + 6421366 p^{3} T^{7} + 452519 p^{6} T^{8} + 26367 p^{9} T^{9} + 1247 p^{12} T^{10} + 43 p^{15} T^{11} + p^{18} T^{12} \) | |
| 11 | \( 1 + 28 T + 3144 T^{2} + 41080 T^{3} + 3977536 T^{4} + 285452 p T^{5} + 4111332998 T^{6} + 285452 p^{4} T^{7} + 3977536 p^{6} T^{8} + 41080 p^{9} T^{9} + 3144 p^{12} T^{10} + 28 p^{15} T^{11} + p^{18} T^{12} \) | |
| 13 | \( 1 + 56 T + 8776 T^{2} + 530884 T^{3} + 37473880 T^{4} + 2150765192 T^{5} + 100690118558 T^{6} + 2150765192 p^{3} T^{7} + 37473880 p^{6} T^{8} + 530884 p^{9} T^{9} + 8776 p^{12} T^{10} + 56 p^{15} T^{11} + p^{18} T^{12} \) | |
| 17 | \( 1 + 19 T + 23143 T^{2} + 543393 T^{3} + 241535186 T^{4} + 5650313095 T^{5} + 1493450611759 T^{6} + 5650313095 p^{3} T^{7} + 241535186 p^{6} T^{8} + 543393 p^{9} T^{9} + 23143 p^{12} T^{10} + 19 p^{15} T^{11} + p^{18} T^{12} \) | |
| 19 | \( 1 - 75 T + 38259 T^{2} - 2365781 T^{3} + 629552155 T^{4} - 31151517862 T^{5} + 5681041321490 T^{6} - 31151517862 p^{3} T^{7} + 629552155 p^{6} T^{8} - 2365781 p^{9} T^{9} + 38259 p^{12} T^{10} - 75 p^{15} T^{11} + p^{18} T^{12} \) | |
| 23 | \( 1 - 131 T + 52195 T^{2} - 3677795 T^{3} + 974730114 T^{4} - 33210519163 T^{5} + 11858751245947 T^{6} - 33210519163 p^{3} T^{7} + 974730114 p^{6} T^{8} - 3677795 p^{9} T^{9} + 52195 p^{12} T^{10} - 131 p^{15} T^{11} + p^{18} T^{12} \) | |
| 29 | \( 1 - 515 T + 204583 T^{2} - 58068807 T^{3} + 13900558631 T^{4} - 2733986934494 T^{5} + 464005745217070 T^{6} - 2733986934494 p^{3} T^{7} + 13900558631 p^{6} T^{8} - 58068807 p^{9} T^{9} + 204583 p^{12} T^{10} - 515 p^{15} T^{11} + p^{18} T^{12} \) | |
| 31 | \( 1 + 237 T + 125373 T^{2} + 21016589 T^{3} + 7321362670 T^{4} + 1031063540341 T^{5} + 275109610824401 T^{6} + 1031063540341 p^{3} T^{7} + 7321362670 p^{6} T^{8} + 21016589 p^{9} T^{9} + 125373 p^{12} T^{10} + 237 p^{15} T^{11} + p^{18} T^{12} \) | |
| 37 | \( 1 - 269 T + 126311 T^{2} - 30748693 T^{3} + 8177560635 T^{4} - 1302357216602 T^{5} + 425119347961066 T^{6} - 1302357216602 p^{3} T^{7} + 8177560635 p^{6} T^{8} - 30748693 p^{9} T^{9} + 126311 p^{12} T^{10} - 269 p^{15} T^{11} + p^{18} T^{12} \) | |
| 41 | \( 1 + 471 T + 349763 T^{2} + 112600045 T^{3} + 50459129866 T^{4} + 12922113800443 T^{5} + 4372223136871043 T^{6} + 12922113800443 p^{3} T^{7} + 50459129866 p^{6} T^{8} + 112600045 p^{9} T^{9} + 349763 p^{12} T^{10} + 471 p^{15} T^{11} + p^{18} T^{12} \) | |
| 47 | \( 1 + 415 T + 421631 T^{2} + 116866317 T^{3} + 77411983523 T^{4} + 16052264514750 T^{5} + 9154052369892234 T^{6} + 16052264514750 p^{3} T^{7} + 77411983523 p^{6} T^{8} + 116866317 p^{9} T^{9} + 421631 p^{12} T^{10} + 415 p^{15} T^{11} + p^{18} T^{12} \) | |
| 53 | \( 1 - 450 T + 321704 T^{2} - 149982378 T^{3} + 77929548632 T^{4} - 28654270442506 T^{5} + 12746558079363422 T^{6} - 28654270442506 p^{3} T^{7} + 77929548632 p^{6} T^{8} - 149982378 p^{9} T^{9} + 321704 p^{12} T^{10} - 450 p^{15} T^{11} + p^{18} T^{12} \) | |
| 59 | \( 1 + 356 T + 1153950 T^{2} + 347607228 T^{3} + 570632518215 T^{4} + 139273869185096 T^{5} + 154351054402988548 T^{6} + 139273869185096 p^{3} T^{7} + 570632518215 p^{6} T^{8} + 347607228 p^{9} T^{9} + 1153950 p^{12} T^{10} + 356 p^{15} T^{11} + p^{18} T^{12} \) | |
| 61 | \( 1 - 1328 T + 1795994 T^{2} - 1454429624 T^{3} + 1136828745699 T^{4} - 649067768079368 T^{5} + 354455789917301172 T^{6} - 649067768079368 p^{3} T^{7} + 1136828745699 p^{6} T^{8} - 1454429624 p^{9} T^{9} + 1795994 p^{12} T^{10} - 1328 p^{15} T^{11} + p^{18} T^{12} \) | |
| 67 | \( 1 + 632 T + 927628 T^{2} + 253029932 T^{3} + 179674044568 T^{4} - 62778009874096 T^{5} - 5212390617577006 T^{6} - 62778009874096 p^{3} T^{7} + 179674044568 p^{6} T^{8} + 253029932 p^{9} T^{9} + 927628 p^{12} T^{10} + 632 p^{15} T^{11} + p^{18} T^{12} \) | |
| 71 | \( 1 + 144 T + 863230 T^{2} + 72744496 T^{3} + 475313592223 T^{4} + 62333254020128 T^{5} + 209643801201434276 T^{6} + 62333254020128 p^{3} T^{7} + 475313592223 p^{6} T^{8} + 72744496 p^{9} T^{9} + 863230 p^{12} T^{10} + 144 p^{15} T^{11} + p^{18} T^{12} \) | |
| 73 | \( 1 + 864 T + 1080658 T^{2} + 439706488 T^{3} + 458263245867 T^{4} + 209583356925592 T^{5} + 222637509631027524 T^{6} + 209583356925592 p^{3} T^{7} + 458263245867 p^{6} T^{8} + 439706488 p^{9} T^{9} + 1080658 p^{12} T^{10} + 864 p^{15} T^{11} + p^{18} T^{12} \) | |
| 79 | \( 1 + 1613 T + 2731081 T^{2} + 3130876751 T^{3} + 3268965374139 T^{4} + 2799662350208018 T^{5} + 2111366737833161318 T^{6} + 2799662350208018 p^{3} T^{7} + 3268965374139 p^{6} T^{8} + 3130876751 p^{9} T^{9} + 2731081 p^{12} T^{10} + 1613 p^{15} T^{11} + p^{18} T^{12} \) | |
| 83 | \( 1 - 682 T + 1120004 T^{2} - 1783287782 T^{3} + 1291656570448 T^{4} - 1149121488958882 T^{5} + 1227368803599345682 T^{6} - 1149121488958882 p^{3} T^{7} + 1291656570448 p^{6} T^{8} - 1783287782 p^{9} T^{9} + 1120004 p^{12} T^{10} - 682 p^{15} T^{11} + p^{18} T^{12} \) | |
| 89 | \( 1 + 3378 T + 7851850 T^{2} + 13055260850 T^{3} + 17370332054203 T^{4} + 19017874978895668 T^{5} + 17369747195795800052 T^{6} + 19017874978895668 p^{3} T^{7} + 17370332054203 p^{6} T^{8} + 13055260850 p^{9} T^{9} + 7851850 p^{12} T^{10} + 3378 p^{15} T^{11} + p^{18} T^{12} \) | |
| 97 | \( 1 - 55 T + 2496871 T^{2} + 940317011 T^{3} + 3317883854770 T^{4} + 1706175158728421 T^{5} + 3579842987764450575 T^{6} + 1706175158728421 p^{3} T^{7} + 3317883854770 p^{6} T^{8} + 940317011 p^{9} T^{9} + 2496871 p^{12} T^{10} - 55 p^{15} T^{11} + p^{18} T^{12} \) | |
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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
−4.73665426294616462440743730271, −4.71462964252738789242401100109, −4.42226840643435072393778447782, −4.34919698070238007852896829360, −4.27890659640130035076934189363, −4.19052920162380503893378837376, −4.10302860128829376418783998507, −3.71507122664241283602017310831, −3.60609564133083975518241830315, −3.52754931245059481400631823647, −3.49355427217738791691096423254, −3.40688191447122841853565940376, −3.02735685591279809657300803777, −2.87092981202235015404603391496, −2.78503292991922820856789559445, −2.63775764402894600203787722337, −2.43637842230234073353911462116, −2.36445395032006644963911844668, −2.02218413275887203851588778566, −1.60295937312262375707562144412, −1.57191331212374278375954778261, −1.13637825556005290828442372407, −1.01705447276979381840415802657, −0.77596027391687280571100104136, −0.74657448352636802478044855011, 0, 0, 0, 0, 0, 0, 0.74657448352636802478044855011, 0.77596027391687280571100104136, 1.01705447276979381840415802657, 1.13637825556005290828442372407, 1.57191331212374278375954778261, 1.60295937312262375707562144412, 2.02218413275887203851588778566, 2.36445395032006644963911844668, 2.43637842230234073353911462116, 2.63775764402894600203787722337, 2.78503292991922820856789559445, 2.87092981202235015404603391496, 3.02735685591279809657300803777, 3.40688191447122841853565940376, 3.49355427217738791691096423254, 3.52754931245059481400631823647, 3.60609564133083975518241830315, 3.71507122664241283602017310831, 4.10302860128829376418783998507, 4.19052920162380503893378837376, 4.27890659640130035076934189363, 4.34919698070238007852896829360, 4.42226840643435072393778447782, 4.71462964252738789242401100109, 4.73665426294616462440743730271
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.