Dirichlet series
| L(s) = 1 | − 6·3-s + 7·4-s − 6·5-s − 3·7-s + 15·9-s − 18·11-s − 42·12-s − 13-s + 36·15-s + 18·16-s + 9·19-s − 42·20-s + 18·21-s + 32·23-s + 8·25-s − 14·27-s − 21·28-s − 5·29-s − 15·31-s + 108·33-s + 18·35-s + 105·36-s + 6·39-s − 15·41-s − 13·43-s − 126·44-s − 90·45-s + ⋯ |
| L(s) = 1 | − 3.46·3-s + 7/2·4-s − 2.68·5-s − 1.13·7-s + 5·9-s − 5.42·11-s − 12.1·12-s − 0.277·13-s + 9.29·15-s + 9/2·16-s + 2.06·19-s − 9.39·20-s + 3.92·21-s + 6.67·23-s + 8/5·25-s − 2.69·27-s − 3.96·28-s − 0.928·29-s − 2.69·31-s + 18.8·33-s + 3.04·35-s + 35/2·36-s + 0.960·39-s − 2.34·41-s − 1.98·43-s − 18.9·44-s − 13.4·45-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(3^{12} \cdot 7^{12} \cdot 13^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(11515.3\) |
| Root analytic conductor: | \(1.47645\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 3^{12} \cdot 7^{12} \cdot 13^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.1455235463\) |
| \(L(\frac12)\) | \(\approx\) | \(0.1455235463\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 + T + T^{2} )^{6} \) |
| 7 | \( 1 + 3 T + 6 T^{2} + 15 T^{3} - 39 T^{4} - 258 T^{5} - 475 T^{6} - 258 p T^{7} - 39 p^{2} T^{8} + 15 p^{3} T^{9} + 6 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) | |
| 13 | \( 1 + T - 2 T^{2} + 28 T^{3} - 34 T^{4} - 575 T^{5} - 1057 T^{6} - 575 p T^{7} - 34 p^{2} T^{8} + 28 p^{3} T^{9} - 2 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \) | |
| good | 2 | \( ( 1 - 3 T + T^{2} + 5 p T^{3} - 15 T^{4} - 5 p T^{5} + 45 T^{6} - 5 p^{2} T^{7} - 15 p^{2} T^{8} + 5 p^{4} T^{9} + p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} )( 1 + 3 T + T^{2} - 5 p T^{3} - 15 T^{4} + 5 p T^{5} + 45 T^{6} + 5 p^{2} T^{7} - 15 p^{2} T^{8} - 5 p^{4} T^{9} + p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} ) \) |
| 5 | \( 1 + 6 T + 28 T^{2} + 96 T^{3} + 287 T^{4} + 771 T^{5} + 367 p T^{6} + 4134 T^{7} + 8591 T^{8} + 17673 T^{9} + 36333 T^{10} + 72132 T^{11} + 161199 T^{12} + 72132 p T^{13} + 36333 p^{2} T^{14} + 17673 p^{3} T^{15} + 8591 p^{4} T^{16} + 4134 p^{5} T^{17} + 367 p^{7} T^{18} + 771 p^{7} T^{19} + 287 p^{8} T^{20} + 96 p^{9} T^{21} + 28 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 + 18 T + 211 T^{2} + 1854 T^{3} + 13598 T^{4} + 85935 T^{5} + 483416 T^{6} + 2452803 T^{7} + 11388431 T^{8} + 48681126 T^{9} + 192966846 T^{10} + 710917434 T^{11} + 2442320775 T^{12} + 710917434 p T^{13} + 192966846 p^{2} T^{14} + 48681126 p^{3} T^{15} + 11388431 p^{4} T^{16} + 2452803 p^{5} T^{17} + 483416 p^{6} T^{18} + 85935 p^{7} T^{19} + 13598 p^{8} T^{20} + 1854 p^{9} T^{21} + 211 p^{10} T^{22} + 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( ( 1 + 91 T^{2} - 4 T^{3} + 3615 T^{4} - 182 T^{5} + 80139 T^{6} - 182 p T^{7} + 3615 p^{2} T^{8} - 4 p^{3} T^{9} + 91 p^{4} T^{10} + p^{6} T^{12} )^{2} \) | |
| 19 | \( 1 - 9 T + 107 T^{2} - 720 T^{3} + 5688 T^{4} - 33246 T^{5} + 206623 T^{6} - 1089774 T^{7} + 5790446 T^{8} - 28400265 T^{9} + 134301546 T^{10} - 32634234 p T^{11} + 142573519 p T^{12} - 32634234 p^{2} T^{13} + 134301546 p^{2} T^{14} - 28400265 p^{3} T^{15} + 5790446 p^{4} T^{16} - 1089774 p^{5} T^{17} + 206623 p^{6} T^{18} - 33246 p^{7} T^{19} + 5688 p^{8} T^{20} - 720 p^{9} T^{21} + 107 p^{10} T^{22} - 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( ( 1 - 16 T + 225 T^{2} - 1966 T^{3} + 15494 T^{4} - 91659 T^{5} + 497509 T^{6} - 91659 p T^{7} + 15494 p^{2} T^{8} - 1966 p^{3} T^{9} + 225 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 29 | \( 1 + 5 T - 109 T^{2} - 444 T^{3} + 6890 T^{4} + 19752 T^{5} - 332151 T^{6} - 544134 T^{7} + 13494374 T^{8} + 9374749 T^{9} - 480329384 T^{10} - 84394380 T^{11} + 14899983267 T^{12} - 84394380 p T^{13} - 480329384 p^{2} T^{14} + 9374749 p^{3} T^{15} + 13494374 p^{4} T^{16} - 544134 p^{5} T^{17} - 332151 p^{6} T^{18} + 19752 p^{7} T^{19} + 6890 p^{8} T^{20} - 444 p^{9} T^{21} - 109 p^{10} T^{22} + 5 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 + 15 T + 223 T^{2} + 2220 T^{3} + 20663 T^{4} + 163659 T^{5} + 1232540 T^{6} + 8596356 T^{7} + 57718679 T^{8} + 369903624 T^{9} + 2275653561 T^{10} + 13491540564 T^{11} + 76411880751 T^{12} + 13491540564 p T^{13} + 2275653561 p^{2} T^{14} + 369903624 p^{3} T^{15} + 57718679 p^{4} T^{16} + 8596356 p^{5} T^{17} + 1232540 p^{6} T^{18} + 163659 p^{7} T^{19} + 20663 p^{8} T^{20} + 2220 p^{9} T^{21} + 223 p^{10} T^{22} + 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 - 168 T^{2} + 16450 T^{4} - 1170523 T^{6} + 67275017 T^{8} - 3196993541 T^{10} + 128572712441 T^{12} - 3196993541 p^{2} T^{14} + 67275017 p^{4} T^{16} - 1170523 p^{6} T^{18} + 16450 p^{8} T^{20} - 168 p^{10} T^{22} + p^{12} T^{24} \) | |
| 41 | \( 1 + 15 T + 174 T^{2} + 1485 T^{3} + 10150 T^{4} + 60999 T^{5} + 361229 T^{6} + 1794345 T^{7} + 7659437 T^{8} + 10158996 T^{9} - 230773790 T^{10} - 3141171528 T^{11} - 22641210097 T^{12} - 3141171528 p T^{13} - 230773790 p^{2} T^{14} + 10158996 p^{3} T^{15} + 7659437 p^{4} T^{16} + 1794345 p^{5} T^{17} + 361229 p^{6} T^{18} + 60999 p^{7} T^{19} + 10150 p^{8} T^{20} + 1485 p^{9} T^{21} + 174 p^{10} T^{22} + 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 + 13 T - 112 T^{2} - 1205 T^{3} + 17787 T^{4} + 102366 T^{5} - 1551405 T^{6} - 4752161 T^{7} + 105630248 T^{8} + 136229581 T^{9} - 5872978169 T^{10} - 2866338304 T^{11} + 261589114081 T^{12} - 2866338304 p T^{13} - 5872978169 p^{2} T^{14} + 136229581 p^{3} T^{15} + 105630248 p^{4} T^{16} - 4752161 p^{5} T^{17} - 1551405 p^{6} T^{18} + 102366 p^{7} T^{19} + 17787 p^{8} T^{20} - 1205 p^{9} T^{21} - 112 p^{10} T^{22} + 13 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 - 9 T + 157 T^{2} - 1170 T^{3} + 10378 T^{4} - 82071 T^{5} + 436039 T^{6} - 3923814 T^{7} + 18883118 T^{8} - 160060026 T^{9} + 1278934561 T^{10} - 6962440686 T^{11} + 74098937369 T^{12} - 6962440686 p T^{13} + 1278934561 p^{2} T^{14} - 160060026 p^{3} T^{15} + 18883118 p^{4} T^{16} - 3923814 p^{5} T^{17} + 436039 p^{6} T^{18} - 82071 p^{7} T^{19} + 10378 p^{8} T^{20} - 1170 p^{9} T^{21} + 157 p^{10} T^{22} - 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 - 18 T - 45 T^{2} + 1062 T^{3} + 22149 T^{4} - 146817 T^{5} - 1791976 T^{6} + 3767067 T^{7} + 139825809 T^{8} + 34875819 T^{9} - 8335105776 T^{10} + 4379744871 T^{11} + 336413841081 T^{12} + 4379744871 p T^{13} - 8335105776 p^{2} T^{14} + 34875819 p^{3} T^{15} + 139825809 p^{4} T^{16} + 3767067 p^{5} T^{17} - 1791976 p^{6} T^{18} - 146817 p^{7} T^{19} + 22149 p^{8} T^{20} + 1062 p^{9} T^{21} - 45 p^{10} T^{22} - 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 - 415 T^{2} + 85321 T^{4} - 11639635 T^{6} + 1186480133 T^{8} - 95649757147 T^{10} + 6248813413337 T^{12} - 95649757147 p^{2} T^{14} + 1186480133 p^{4} T^{16} - 11639635 p^{6} T^{18} + 85321 p^{8} T^{20} - 415 p^{10} T^{22} + p^{12} T^{24} \) | |
| 61 | \( 1 - 26 T + 287 T^{2} - 1400 T^{3} - 4188 T^{4} + 133593 T^{5} - 1084188 T^{6} + 2575405 T^{7} + 677189 p T^{8} - 588034958 T^{9} + 3683610724 T^{10} - 4161224764 T^{11} - 81054085541 T^{12} - 4161224764 p T^{13} + 3683610724 p^{2} T^{14} - 588034958 p^{3} T^{15} + 677189 p^{5} T^{16} + 2575405 p^{5} T^{17} - 1084188 p^{6} T^{18} + 133593 p^{7} T^{19} - 4188 p^{8} T^{20} - 1400 p^{9} T^{21} + 287 p^{10} T^{22} - 26 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 + 24 T + 457 T^{2} + 6360 T^{3} + 71208 T^{4} + 737361 T^{5} + 7140470 T^{6} + 68537973 T^{7} + 670090103 T^{8} + 5996139708 T^{9} + 51750851640 T^{10} + 425147105238 T^{11} + 3399818384923 T^{12} + 425147105238 p T^{13} + 51750851640 p^{2} T^{14} + 5996139708 p^{3} T^{15} + 670090103 p^{4} T^{16} + 68537973 p^{5} T^{17} + 7140470 p^{6} T^{18} + 737361 p^{7} T^{19} + 71208 p^{8} T^{20} + 6360 p^{9} T^{21} + 457 p^{10} T^{22} + 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 + 15 T + 289 T^{2} + 3210 T^{3} + 36304 T^{4} + 306894 T^{5} + 2262055 T^{6} + 14678646 T^{7} + 61717448 T^{8} + 318907011 T^{9} + 326960200 T^{10} + 5121702606 T^{11} - 30333999439 T^{12} + 5121702606 p T^{13} + 326960200 p^{2} T^{14} + 318907011 p^{3} T^{15} + 61717448 p^{4} T^{16} + 14678646 p^{5} T^{17} + 2262055 p^{6} T^{18} + 306894 p^{7} T^{19} + 36304 p^{8} T^{20} + 3210 p^{9} T^{21} + 289 p^{10} T^{22} + 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 - 18 T + 376 T^{2} - 4824 T^{3} + 62850 T^{4} - 818526 T^{5} + 9206366 T^{6} - 109179408 T^{7} + 1065941066 T^{8} - 10625515272 T^{9} + 101733004842 T^{10} - 911693353392 T^{11} + 8417624065063 T^{12} - 911693353392 p T^{13} + 101733004842 p^{2} T^{14} - 10625515272 p^{3} T^{15} + 1065941066 p^{4} T^{16} - 109179408 p^{5} T^{17} + 9206366 p^{6} T^{18} - 818526 p^{7} T^{19} + 62850 p^{8} T^{20} - 4824 p^{9} T^{21} + 376 p^{10} T^{22} - 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 + 4 T - 330 T^{2} + 176 T^{3} + 65125 T^{4} - 184808 T^{5} - 7944766 T^{6} + 36716796 T^{7} + 695397274 T^{8} - 3357835540 T^{9} - 46538295314 T^{10} + 125713245864 T^{11} + 3293869827229 T^{12} + 125713245864 p T^{13} - 46538295314 p^{2} T^{14} - 3357835540 p^{3} T^{15} + 695397274 p^{4} T^{16} + 36716796 p^{5} T^{17} - 7944766 p^{6} T^{18} - 184808 p^{7} T^{19} + 65125 p^{8} T^{20} + 176 p^{9} T^{21} - 330 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 - 648 T^{2} + 198490 T^{4} - 38405290 T^{6} + 5339197007 T^{8} - 580215727859 T^{10} + 52282941288107 T^{12} - 580215727859 p^{2} T^{14} + 5339197007 p^{4} T^{16} - 38405290 p^{6} T^{18} + 198490 p^{8} T^{20} - 648 p^{10} T^{22} + p^{12} T^{24} \) | |
| 89 | \( 1 - 758 T^{2} + 282739 T^{4} - 68209313 T^{6} + 11810101205 T^{8} - 1542553311626 T^{10} + 155643549309737 T^{12} - 1542553311626 p^{2} T^{14} + 11810101205 p^{4} T^{16} - 68209313 p^{6} T^{18} + 282739 p^{8} T^{20} - 758 p^{10} T^{22} + p^{12} T^{24} \) | |
| 97 | \( 1 - 45 T + 1267 T^{2} - 26640 T^{3} + 449075 T^{4} - 6432468 T^{5} + 81761366 T^{6} - 956423301 T^{7} + 10753777628 T^{8} - 118717850235 T^{9} + 1291993782855 T^{10} - 13662139782249 T^{11} + 138119510963013 T^{12} - 13662139782249 p T^{13} + 1291993782855 p^{2} T^{14} - 118717850235 p^{3} T^{15} + 10753777628 p^{4} T^{16} - 956423301 p^{5} T^{17} + 81761366 p^{6} T^{18} - 6432468 p^{7} T^{19} + 449075 p^{8} T^{20} - 26640 p^{9} T^{21} + 1267 p^{10} T^{22} - 45 p^{11} T^{23} + p^{12} T^{24} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.12196069363936998397305568785, −4.07941173726345471382767858444, −3.68481388264130666169483402917, −3.56331549850554283527676113237, −3.53283429474610297983902498357, −3.42258882623789690383900363650, −3.37109795587309443833146615131, −3.27507892252319340599683261475, −3.15230279640586358210394432517, −3.05936861474093561404332687224, −3.00976263376932620294718525368, −2.88112555195610081689085783149, −2.53147912316014446095718695361, −2.51903289299371847162521829748, −2.34560143341164224847934527528, −2.27862732423982445946330831060, −2.21142471867404603474950576375, −2.18057711387004243371374396968, −1.78553276445211003005052734010, −1.47933175597452228431396741243, −1.36301522957306252984891587070, −0.899299185338644451820390029781, −0.820742982557997120856144643589, −0.31384944671436732298957379366, −0.29456389191215108880550766482, 0.29456389191215108880550766482, 0.31384944671436732298957379366, 0.820742982557997120856144643589, 0.899299185338644451820390029781, 1.36301522957306252984891587070, 1.47933175597452228431396741243, 1.78553276445211003005052734010, 2.18057711387004243371374396968, 2.21142471867404603474950576375, 2.27862732423982445946330831060, 2.34560143341164224847934527528, 2.51903289299371847162521829748, 2.53147912316014446095718695361, 2.88112555195610081689085783149, 3.00976263376932620294718525368, 3.05936861474093561404332687224, 3.15230279640586358210394432517, 3.27507892252319340599683261475, 3.37109795587309443833146615131, 3.42258882623789690383900363650, 3.53283429474610297983902498357, 3.56331549850554283527676113237, 3.68481388264130666169483402917, 4.07941173726345471382767858444, 4.12196069363936998397305568785
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.