Dirichlet series
| L(s) = 1 | − 2·2-s + 16·3-s − 15·4-s − 30·5-s − 32·6-s + 16·8-s + 82·9-s + 60·10-s − 16·11-s − 240·12-s + 168·13-s − 480·15-s + 191·16-s − 4·17-s − 164·18-s + 308·19-s + 450·20-s + 32·22-s − 336·23-s + 256·24-s + 525·25-s − 336·26-s + 124·27-s + 176·29-s + 960·30-s + 392·31-s − 158·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 3.07·3-s − 1.87·4-s − 2.68·5-s − 2.17·6-s + 0.707·8-s + 3.03·9-s + 1.89·10-s − 0.438·11-s − 5.77·12-s + 3.58·13-s − 8.26·15-s + 2.98·16-s − 0.0570·17-s − 2.14·18-s + 3.71·19-s + 5.03·20-s + 0.310·22-s − 3.04·23-s + 2.17·24-s + 21/5·25-s − 2.53·26-s + 0.883·27-s + 1.12·29-s + 5.84·30-s + 2.27·31-s − 0.872·32-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(5^{6} \cdot 7^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(9.12416\times 10^{6}\) |
| Root analytic conductor: | \(3.80203\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 5^{6} \cdot 7^{12} ,\ ( \ : [3/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(6.642102582\) |
| \(L(\frac12)\) | \(\approx\) | \(6.642102582\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( ( 1 + p T )^{6} \) |
| 7 | \( 1 \) | |
| good | 2 | \( 1 + p T + 19 T^{2} + 13 p^{2} T^{3} + 83 p T^{4} + 73 p^{3} T^{5} + 305 p^{2} T^{6} + 73 p^{6} T^{7} + 83 p^{7} T^{8} + 13 p^{11} T^{9} + 19 p^{12} T^{10} + p^{16} T^{11} + p^{18} T^{12} \) |
| 3 | \( 1 - 16 T + 58 p T^{2} - 532 p T^{3} + 11876 T^{4} - 2800 p^{3} T^{5} + 425492 T^{6} - 2800 p^{6} T^{7} + 11876 p^{6} T^{8} - 532 p^{10} T^{9} + 58 p^{13} T^{10} - 16 p^{15} T^{11} + p^{18} T^{12} \) | |
| 11 | \( 1 + 16 T + 812 T^{2} - 15300 T^{3} + 2567184 T^{4} - 56858680 T^{5} - 552711974 T^{6} - 56858680 p^{3} T^{7} + 2567184 p^{6} T^{8} - 15300 p^{9} T^{9} + 812 p^{12} T^{10} + 16 p^{15} T^{11} + p^{18} T^{12} \) | |
| 13 | \( 1 - 168 T + 18956 T^{2} - 116156 p T^{3} + 104466904 T^{4} - 6040563440 T^{5} + 309756529486 T^{6} - 6040563440 p^{3} T^{7} + 104466904 p^{6} T^{8} - 116156 p^{10} T^{9} + 18956 p^{12} T^{10} - 168 p^{15} T^{11} + p^{18} T^{12} \) | |
| 17 | \( 1 + 4 T + 14086 T^{2} - 160336 T^{3} + 115923452 T^{4} - 1228657188 T^{5} + 692428906716 T^{6} - 1228657188 p^{3} T^{7} + 115923452 p^{6} T^{8} - 160336 p^{9} T^{9} + 14086 p^{12} T^{10} + 4 p^{15} T^{11} + p^{18} T^{12} \) | |
| 19 | \( 1 - 308 T + 73404 T^{2} - 12077372 T^{3} + 1625488783 T^{4} - 176451707096 T^{5} + 16038736148248 T^{6} - 176451707096 p^{3} T^{7} + 1625488783 p^{6} T^{8} - 12077372 p^{9} T^{9} + 73404 p^{12} T^{10} - 308 p^{15} T^{11} + p^{18} T^{12} \) | |
| 23 | \( 1 + 336 T + 98630 T^{2} + 19185056 T^{3} + 3349928211 T^{4} + 19757853136 p T^{5} + 105705477964 p^{2} T^{6} + 19757853136 p^{4} T^{7} + 3349928211 p^{6} T^{8} + 19185056 p^{9} T^{9} + 98630 p^{12} T^{10} + 336 p^{15} T^{11} + p^{18} T^{12} \) | |
| 29 | \( 1 - 176 T + 92656 T^{2} - 12546380 T^{3} + 4288608312 T^{4} - 478053343680 T^{5} + 127431913139622 T^{6} - 478053343680 p^{3} T^{7} + 4288608312 p^{6} T^{8} - 12546380 p^{9} T^{9} + 92656 p^{12} T^{10} - 176 p^{15} T^{11} + p^{18} T^{12} \) | |
| 31 | \( 1 - 392 T + 153030 T^{2} - 32287912 T^{3} + 7677507203 T^{4} - 1249077738464 T^{5} + 255230995151820 T^{6} - 1249077738464 p^{3} T^{7} + 7677507203 p^{6} T^{8} - 32287912 p^{9} T^{9} + 153030 p^{12} T^{10} - 392 p^{15} T^{11} + p^{18} T^{12} \) | |
| 37 | \( 1 + 140 T + 213942 T^{2} + 12843948 T^{3} + 18729025499 T^{4} + 132721017192 T^{5} + 1059724116139756 T^{6} + 132721017192 p^{3} T^{7} + 18729025499 p^{6} T^{8} + 12843948 p^{9} T^{9} + 213942 p^{12} T^{10} + 140 p^{15} T^{11} + p^{18} T^{12} \) | |
| 41 | \( 1 - 16 p T + 421976 T^{2} - 166159344 T^{3} + 65879968003 T^{4} - 19715169399936 T^{5} + 5871622216065200 T^{6} - 19715169399936 p^{3} T^{7} + 65879968003 p^{6} T^{8} - 166159344 p^{9} T^{9} + 421976 p^{12} T^{10} - 16 p^{16} T^{11} + p^{18} T^{12} \) | |
| 43 | \( 1 + 388 T + 421158 T^{2} + 116550300 T^{3} + 73177187783 T^{4} + 15517478343240 T^{5} + 7317237698378164 T^{6} + 15517478343240 p^{3} T^{7} + 73177187783 p^{6} T^{8} + 116550300 p^{9} T^{9} + 421158 p^{12} T^{10} + 388 p^{15} T^{11} + p^{18} T^{12} \) | |
| 47 | \( 1 - 628 T + 589452 T^{2} - 246437960 T^{3} + 139029029008 T^{4} - 44711775028180 T^{5} + 18650987083950686 T^{6} - 44711775028180 p^{3} T^{7} + 139029029008 p^{6} T^{8} - 246437960 p^{9} T^{9} + 589452 p^{12} T^{10} - 628 p^{15} T^{11} + p^{18} T^{12} \) | |
| 53 | \( 1 + 676 T + 756806 T^{2} + 367769844 T^{3} + 249160429851 T^{4} + 95287278237352 T^{5} + 47830207170007020 T^{6} + 95287278237352 p^{3} T^{7} + 249160429851 p^{6} T^{8} + 367769844 p^{9} T^{9} + 756806 p^{12} T^{10} + 676 p^{15} T^{11} + p^{18} T^{12} \) | |
| 59 | \( 1 - 996 T + 1036536 T^{2} - 686090796 T^{3} + 452936806907 T^{4} - 228269279812616 T^{5} + 117358818568794736 T^{6} - 228269279812616 p^{3} T^{7} + 452936806907 p^{6} T^{8} - 686090796 p^{9} T^{9} + 1036536 p^{12} T^{10} - 996 p^{15} T^{11} + p^{18} T^{12} \) | |
| 61 | \( 1 - 740 T + 970482 T^{2} - 661891780 T^{3} + 495713522631 T^{4} - 257223694985640 T^{5} + 147582202110218748 T^{6} - 257223694985640 p^{3} T^{7} + 495713522631 p^{6} T^{8} - 661891780 p^{9} T^{9} + 970482 p^{12} T^{10} - 740 p^{15} T^{11} + p^{18} T^{12} \) | |
| 67 | \( 1 - 1768 T + 2799902 T^{2} - 2855569976 T^{3} + 2569585021227 T^{4} - 1777311287531536 T^{5} + 1093536943175084732 T^{6} - 1777311287531536 p^{3} T^{7} + 2569585021227 p^{6} T^{8} - 2855569976 p^{9} T^{9} + 2799902 p^{12} T^{10} - 1768 p^{15} T^{11} + p^{18} T^{12} \) | |
| 71 | \( 1 + 224 T + 735802 T^{2} - 33746048 T^{3} + 362779099663 T^{4} + 5782399589792 T^{5} + 175695286389376652 T^{6} + 5782399589792 p^{3} T^{7} + 362779099663 p^{6} T^{8} - 33746048 p^{9} T^{9} + 735802 p^{12} T^{10} + 224 p^{15} T^{11} + p^{18} T^{12} \) | |
| 73 | \( 1 - 2640 T + 4127428 T^{2} - 4281599440 T^{3} + 3477156414455 T^{4} - 2335723050987360 T^{5} + 1487916107494354120 T^{6} - 2335723050987360 p^{3} T^{7} + 3477156414455 p^{6} T^{8} - 4281599440 p^{9} T^{9} + 4127428 p^{12} T^{10} - 2640 p^{15} T^{11} + p^{18} T^{12} \) | |
| 79 | \( 1 - 1636 T + 3307008 T^{2} - 41764072 p T^{3} + 3796696094800 T^{4} - 2753755692366396 T^{5} + 2363360511013187334 T^{6} - 2753755692366396 p^{3} T^{7} + 3796696094800 p^{6} T^{8} - 41764072 p^{10} T^{9} + 3307008 p^{12} T^{10} - 1636 p^{15} T^{11} + p^{18} T^{12} \) | |
| 83 | \( 1 - 140 T + 1262996 T^{2} + 466318380 T^{3} + 677854086191 T^{4} + 649780965410440 T^{5} + 346141233175430792 T^{6} + 649780965410440 p^{3} T^{7} + 677854086191 p^{6} T^{8} + 466318380 p^{9} T^{9} + 1262996 p^{12} T^{10} - 140 p^{15} T^{11} + p^{18} T^{12} \) | |
| 89 | \( 1 + 1904 T + 3824692 T^{2} + 4929406096 T^{3} + 6160110405943 T^{4} + 5994022973389568 T^{5} + 5629689575835156904 T^{6} + 5994022973389568 p^{3} T^{7} + 6160110405943 p^{6} T^{8} + 4929406096 p^{9} T^{9} + 3824692 p^{12} T^{10} + 1904 p^{15} T^{11} + p^{18} T^{12} \) | |
| 97 | \( 1 - 516 T + 3952326 T^{2} - 925377376 T^{3} + 6580438313612 T^{4} - 355370807466588 T^{5} + 6947174177634861996 T^{6} - 355370807466588 p^{3} T^{7} + 6580438313612 p^{6} T^{8} - 925377376 p^{9} T^{9} + 3952326 p^{12} T^{10} - 516 p^{15} T^{11} + p^{18} T^{12} \) | |
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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
−6.21176757043133334540673196895, −6.17974204205319304445565717980, −5.55297317324436657556505914879, −5.44783445946901084243456785488, −5.39055546278072204146151990897, −5.08815908510236258722617345375, −5.04380128049639869261488655691, −4.45671765517778703879461038467, −4.42665986229446890989578114942, −3.96455927600963374563534105028, −3.93172804780944131526303425469, −3.86057040794994841835830471653, −3.64768262180385344804051248003, −3.43722742518968477277897081220, −3.16501718602807753075128091919, −3.12272069389520600421644616893, −3.09527119160272334310761165805, −2.51194248576325288467810013896, −2.35135796975497122965282162208, −2.16401220417760296020271095160, −1.28527926995927466140652778978, −0.984309233975212929154759035161, −0.893047583937501532194086972774, −0.61769456582050065369211343679, −0.45982755472356528798824533563, 0.45982755472356528798824533563, 0.61769456582050065369211343679, 0.893047583937501532194086972774, 0.984309233975212929154759035161, 1.28527926995927466140652778978, 2.16401220417760296020271095160, 2.35135796975497122965282162208, 2.51194248576325288467810013896, 3.09527119160272334310761165805, 3.12272069389520600421644616893, 3.16501718602807753075128091919, 3.43722742518968477277897081220, 3.64768262180385344804051248003, 3.86057040794994841835830471653, 3.93172804780944131526303425469, 3.96455927600963374563534105028, 4.42665986229446890989578114942, 4.45671765517778703879461038467, 5.04380128049639869261488655691, 5.08815908510236258722617345375, 5.39055546278072204146151990897, 5.44783445946901084243456785488, 5.55297317324436657556505914879, 6.17974204205319304445565717980, 6.21176757043133334540673196895
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.