Dirichlet series
| L(s) = 1 | + 7·3-s + 3·5-s + 7·7-s − 41·9-s + 72·11-s + 15·13-s + 21·15-s + 83·19-s + 49·21-s + 141·23-s − 239·25-s − 433·27-s − 249·29-s + 106·31-s + 504·33-s + 21·35-s + 170·37-s + 105·39-s − 100·41-s − 90·43-s − 123·45-s + 372·47-s − 843·49-s − 23·53-s + 216·55-s + 581·57-s + 784·59-s + ⋯ |
| L(s) = 1 | + 1.34·3-s + 0.268·5-s + 0.377·7-s − 1.51·9-s + 1.97·11-s + 0.320·13-s + 0.361·15-s + 1.00·19-s + 0.509·21-s + 1.27·23-s − 1.91·25-s − 3.08·27-s − 1.59·29-s + 0.614·31-s + 2.65·33-s + 0.101·35-s + 0.755·37-s + 0.431·39-s − 0.380·41-s − 0.319·43-s − 0.407·45-s + 1.15·47-s − 2.45·49-s − 0.0596·53-s + 0.529·55-s + 1.35·57-s + 1.72·59-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 17^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 17^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(2^{18} \cdot 17^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.44352\times 10^{12}\) |
| Root analytic conductor: | \(11.6795\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 2^{18} \cdot 17^{12} ,\ ( \ : [3/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(26.81846334\) |
| \(L(\frac12)\) | \(\approx\) | \(26.81846334\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 \) | |
| good | 3 | \( 1 - 7 T + 10 p^{2} T^{2} - 484 T^{3} + 487 p^{2} T^{4} - 20776 T^{5} + 150047 T^{6} - 20776 p^{3} T^{7} + 487 p^{8} T^{8} - 484 p^{9} T^{9} + 10 p^{14} T^{10} - 7 p^{15} T^{11} + p^{18} T^{12} \) |
| 5 | \( 1 - 3 T + 248 T^{2} + 734 T^{3} + 8541 p T^{4} + 50598 p T^{5} + 5033769 T^{6} + 50598 p^{4} T^{7} + 8541 p^{7} T^{8} + 734 p^{9} T^{9} + 248 p^{12} T^{10} - 3 p^{15} T^{11} + p^{18} T^{12} \) | |
| 7 | \( 1 - p T + 892 T^{2} - 284 p^{2} T^{3} + 408257 T^{4} - 9863144 T^{5} + 139269605 T^{6} - 9863144 p^{3} T^{7} + 408257 p^{6} T^{8} - 284 p^{11} T^{9} + 892 p^{12} T^{10} - p^{16} T^{11} + p^{18} T^{12} \) | |
| 11 | \( 1 - 72 T + 529 p T^{2} - 307784 T^{3} + 15027806 T^{4} - 630562184 T^{5} + 2179565005 p T^{6} - 630562184 p^{3} T^{7} + 15027806 p^{6} T^{8} - 307784 p^{9} T^{9} + 529 p^{13} T^{10} - 72 p^{15} T^{11} + p^{18} T^{12} \) | |
| 13 | \( 1 - 15 T + 7570 T^{2} - 293454 T^{3} + 24574755 T^{4} - 1515951654 T^{5} + 56446305023 T^{6} - 1515951654 p^{3} T^{7} + 24574755 p^{6} T^{8} - 293454 p^{9} T^{9} + 7570 p^{12} T^{10} - 15 p^{15} T^{11} + p^{18} T^{12} \) | |
| 19 | \( 1 - 83 T + 6892 T^{2} - 305000 T^{3} + 71239793 T^{4} - 1318656668 T^{5} + 129092678845 T^{6} - 1318656668 p^{3} T^{7} + 71239793 p^{6} T^{8} - 305000 p^{9} T^{9} + 6892 p^{12} T^{10} - 83 p^{15} T^{11} + p^{18} T^{12} \) | |
| 23 | \( 1 - 141 T + 37838 T^{2} - 3787512 T^{3} + 797162219 T^{4} - 62153564676 T^{5} + 10528753989235 T^{6} - 62153564676 p^{3} T^{7} + 797162219 p^{6} T^{8} - 3787512 p^{9} T^{9} + 37838 p^{12} T^{10} - 141 p^{15} T^{11} + p^{18} T^{12} \) | |
| 29 | \( 1 + 249 T + 130378 T^{2} + 28176850 T^{3} + 7564318379 T^{4} + 1315784573002 T^{5} + 242927074200295 T^{6} + 1315784573002 p^{3} T^{7} + 7564318379 p^{6} T^{8} + 28176850 p^{9} T^{9} + 130378 p^{12} T^{10} + 249 p^{15} T^{11} + p^{18} T^{12} \) | |
| 31 | \( 1 - 106 T + 2919 p T^{2} - 267930 T^{3} + 2499010322 T^{4} + 380707906358 T^{5} + 44041084868265 T^{6} + 380707906358 p^{3} T^{7} + 2499010322 p^{6} T^{8} - 267930 p^{9} T^{9} + 2919 p^{13} T^{10} - 106 p^{15} T^{11} + p^{18} T^{12} \) | |
| 37 | \( 1 - 170 T + 203918 T^{2} - 23617746 T^{3} + 20335659975 T^{4} - 1959107546652 T^{5} + 1294306961492068 T^{6} - 1959107546652 p^{3} T^{7} + 20335659975 p^{6} T^{8} - 23617746 p^{9} T^{9} + 203918 p^{12} T^{10} - 170 p^{15} T^{11} + p^{18} T^{12} \) | |
| 41 | \( 1 + 100 T + 226226 T^{2} + 27648820 T^{3} + 26981094719 T^{4} + 3473585233992 T^{5} + 2186649794509564 T^{6} + 3473585233992 p^{3} T^{7} + 26981094719 p^{6} T^{8} + 27648820 p^{9} T^{9} + 226226 p^{12} T^{10} + 100 p^{15} T^{11} + p^{18} T^{12} \) | |
| 43 | \( 1 + 90 T + 399393 T^{2} + 32234898 T^{3} + 71581957274 T^{4} + 4876535629866 T^{5} + 7334707852958001 T^{6} + 4876535629866 p^{3} T^{7} + 71581957274 p^{6} T^{8} + 32234898 p^{9} T^{9} + 399393 p^{12} T^{10} + 90 p^{15} T^{11} + p^{18} T^{12} \) | |
| 47 | \( 1 - 372 T + 173095 T^{2} - 68449716 T^{3} + 21646025854 T^{4} - 6144754206260 T^{5} + 2939523924184899 T^{6} - 6144754206260 p^{3} T^{7} + 21646025854 p^{6} T^{8} - 68449716 p^{9} T^{9} + 173095 p^{12} T^{10} - 372 p^{15} T^{11} + p^{18} T^{12} \) | |
| 53 | \( 1 + 23 T + 92718 T^{2} - 9861266 T^{3} + 19237186567 T^{4} + 1028507304446 T^{5} + 1001569763407 p T^{6} + 1028507304446 p^{3} T^{7} + 19237186567 p^{6} T^{8} - 9861266 p^{9} T^{9} + 92718 p^{12} T^{10} + 23 p^{15} T^{11} + p^{18} T^{12} \) | |
| 59 | \( 1 - 784 T + 1009686 T^{2} - 563860336 T^{3} + 436367604455 T^{4} - 190031070413344 T^{5} + 112050256603334612 T^{6} - 190031070413344 p^{3} T^{7} + 436367604455 p^{6} T^{8} - 563860336 p^{9} T^{9} + 1009686 p^{12} T^{10} - 784 p^{15} T^{11} + p^{18} T^{12} \) | |
| 61 | \( 1 - 92 T + 793787 T^{2} + 13318140 T^{3} + 335730226542 T^{4} + 8149419565524 T^{5} + 95129496492616495 T^{6} + 8149419565524 p^{3} T^{7} + 335730226542 p^{6} T^{8} + 13318140 p^{9} T^{9} + 793787 p^{12} T^{10} - 92 p^{15} T^{11} + p^{18} T^{12} \) | |
| 67 | \( 1 - 238 T + 965730 T^{2} - 40959578 T^{3} + 426345621303 T^{4} + 35268140827868 T^{5} + 137268159012001532 T^{6} + 35268140827868 p^{3} T^{7} + 426345621303 p^{6} T^{8} - 40959578 p^{9} T^{9} + 965730 p^{12} T^{10} - 238 p^{15} T^{11} + p^{18} T^{12} \) | |
| 71 | \( 1 - 940 T + 2280230 T^{2} - 1625045476 T^{3} + 2099008560127 T^{4} - 1143242292795672 T^{5} + 1010175519841730068 T^{6} - 1143242292795672 p^{3} T^{7} + 2099008560127 p^{6} T^{8} - 1625045476 p^{9} T^{9} + 2280230 p^{12} T^{10} - 940 p^{15} T^{11} + p^{18} T^{12} \) | |
| 73 | \( 1 - 692 T + 1064787 T^{2} - 341446748 T^{3} + 317151519558 T^{4} - 14515289763268 T^{5} + 65514674818087527 T^{6} - 14515289763268 p^{3} T^{7} + 317151519558 p^{6} T^{8} - 341446748 p^{9} T^{9} + 1064787 p^{12} T^{10} - 692 p^{15} T^{11} + p^{18} T^{12} \) | |
| 79 | \( 1 - 84 T + 2384630 T^{2} - 120940028 T^{3} + 2591703230063 T^{4} - 89035656773608 T^{5} + 1634702099346454964 T^{6} - 89035656773608 p^{3} T^{7} + 2591703230063 p^{6} T^{8} - 120940028 p^{9} T^{9} + 2384630 p^{12} T^{10} - 84 p^{15} T^{11} + p^{18} T^{12} \) | |
| 83 | \( 1 - 1393 T + 2841518 T^{2} - 3433401660 T^{3} + 3728899205963 T^{4} - 3652815831967184 T^{5} + 2768261667351382283 T^{6} - 3652815831967184 p^{3} T^{7} + 3728899205963 p^{6} T^{8} - 3433401660 p^{9} T^{9} + 2841518 p^{12} T^{10} - 1393 p^{15} T^{11} + p^{18} T^{12} \) | |
| 89 | \( 1 - 976 T + 3338234 T^{2} - 2462653456 T^{3} + 4826644511871 T^{4} - 2787750727791264 T^{5} + 4188186811139213356 T^{6} - 2787750727791264 p^{3} T^{7} + 4826644511871 p^{6} T^{8} - 2462653456 p^{9} T^{9} + 3338234 p^{12} T^{10} - 976 p^{15} T^{11} + p^{18} T^{12} \) | |
| 97 | \( 1 + 325 T + 47784 T^{2} + 860841506 T^{3} + 1363566165329 T^{4} + 716893937781882 T^{5} + 601673724356017713 T^{6} + 716893937781882 p^{3} T^{7} + 1363566165329 p^{6} T^{8} + 860841506 p^{9} T^{9} + 47784 p^{12} T^{10} + 325 p^{15} T^{11} + p^{18} T^{12} \) | |
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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
−4.32350475511808649402360393889, −3.96944448558246128919441950035, −3.77861663901501026021001470296, −3.77425179202352392119438087673, −3.64252000376916778992966245322, −3.61078904384781141862130912063, −3.50887406791139420352587837881, −3.26704534773327281755647651732, −2.93249459359066468332489374419, −2.80979661995705868045213394695, −2.78563534419685557427275844444, −2.75867655756503150380029383767, −2.50734599099511189980820116985, −2.14872479164971782171423915875, −1.96497839239031374383019909874, −1.93354605482374846546326026117, −1.80987529646889082253428137401, −1.61066336073059060506410205907, −1.50483224388013871332327835057, −1.07374204436425849210210941525, −0.883087987672124297611307542252, −0.802081045809578130587132184424, −0.56580909857743669747762404729, −0.39493962737634904265364730209, −0.25541053492809605091450674549, 0.25541053492809605091450674549, 0.39493962737634904265364730209, 0.56580909857743669747762404729, 0.802081045809578130587132184424, 0.883087987672124297611307542252, 1.07374204436425849210210941525, 1.50483224388013871332327835057, 1.61066336073059060506410205907, 1.80987529646889082253428137401, 1.93354605482374846546326026117, 1.96497839239031374383019909874, 2.14872479164971782171423915875, 2.50734599099511189980820116985, 2.75867655756503150380029383767, 2.78563534419685557427275844444, 2.80979661995705868045213394695, 2.93249459359066468332489374419, 3.26704534773327281755647651732, 3.50887406791139420352587837881, 3.61078904384781141862130912063, 3.64252000376916778992966245322, 3.77425179202352392119438087673, 3.77861663901501026021001470296, 3.96944448558246128919441950035, 4.32350475511808649402360393889
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.