Dirichlet series
| L(s) = 1 | − 7·2-s − 7·3-s + 22·4-s − 5-s + 49·6-s − 11·7-s − 35·8-s + 20·9-s + 7·10-s + 7·11-s − 154·12-s + 9·13-s + 77·14-s + 7·15-s + 11·16-s − 140·18-s − 7·19-s − 22·20-s + 77·21-s − 49·22-s − 5·23-s + 245·24-s + 12·25-s − 63·26-s − 28·27-s − 242·28-s − 15·29-s + ⋯ |
| L(s) = 1 | − 4.94·2-s − 4.04·3-s + 11·4-s − 0.447·5-s + 20.0·6-s − 4.15·7-s − 12.3·8-s + 20/3·9-s + 2.21·10-s + 2.11·11-s − 44.4·12-s + 2.49·13-s + 20.5·14-s + 1.80·15-s + 11/4·16-s − 32.9·18-s − 1.60·19-s − 4.91·20-s + 16.8·21-s − 10.4·22-s − 1.04·23-s + 50.0·24-s + 12/5·25-s − 12.3·26-s − 5.38·27-s − 45.7·28-s − 2.78·29-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(29^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(29^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(29^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.37739\times 10^{-8}\) |
| Root analytic conductor: | \(0.481213\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 29^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.0004437374141\) |
| \(L(\frac12)\) | \(\approx\) | \(0.0004437374141\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 29 | \( 1 + 15 T + 126 T^{2} + 622 T^{3} + 1665 T^{4} - 4109 T^{5} - 52800 T^{6} - 4109 p T^{7} + 1665 p^{2} T^{8} + 622 p^{3} T^{9} + 126 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \) |
| good | 2 | \( 1 + 7 T + 27 T^{2} + 35 p T^{3} + 65 p T^{4} + 161 T^{5} + 21 p^{2} T^{6} - 147 T^{7} - 401 T^{8} - 301 T^{9} + 323 p T^{10} + 1267 p T^{11} + 4577 T^{12} + 1267 p^{2} T^{13} + 323 p^{3} T^{14} - 301 p^{3} T^{15} - 401 p^{4} T^{16} - 147 p^{5} T^{17} + 21 p^{8} T^{18} + 161 p^{7} T^{19} + 65 p^{9} T^{20} + 35 p^{10} T^{21} + 27 p^{10} T^{22} + 7 p^{11} T^{23} + p^{12} T^{24} \) |
| 3 | \( 1 + 7 T + 29 T^{2} + 91 T^{3} + 244 T^{4} + 182 p T^{5} + 1012 T^{6} + 497 p T^{7} + 499 p T^{8} - 259 T^{9} - 5693 T^{10} - 16660 T^{11} - 32900 T^{12} - 16660 p T^{13} - 5693 p^{2} T^{14} - 259 p^{3} T^{15} + 499 p^{5} T^{16} + 497 p^{6} T^{17} + 1012 p^{6} T^{18} + 182 p^{8} T^{19} + 244 p^{8} T^{20} + 91 p^{9} T^{21} + 29 p^{10} T^{22} + 7 p^{11} T^{23} + p^{12} T^{24} \) | |
| 5 | \( 1 + T - 11 T^{2} - 11 T^{3} + 66 T^{4} + 76 T^{5} - 174 T^{6} - 69 T^{7} + 399 T^{8} - 1417 T^{9} - 321 p T^{10} + 228 p^{2} T^{11} + 14536 T^{12} + 228 p^{3} T^{13} - 321 p^{3} T^{14} - 1417 p^{3} T^{15} + 399 p^{4} T^{16} - 69 p^{5} T^{17} - 174 p^{6} T^{18} + 76 p^{7} T^{19} + 66 p^{8} T^{20} - 11 p^{9} T^{21} - 11 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \) | |
| 7 | \( 1 + 11 T + 47 T^{2} + 123 T^{3} + 458 T^{4} + 2166 T^{5} + 1054 p T^{6} + 20319 T^{7} + 61291 T^{8} + 197341 T^{9} + 577023 T^{10} + 1543704 T^{11} + 4058132 T^{12} + 1543704 p T^{13} + 577023 p^{2} T^{14} + 197341 p^{3} T^{15} + 61291 p^{4} T^{16} + 20319 p^{5} T^{17} + 1054 p^{7} T^{18} + 2166 p^{7} T^{19} + 458 p^{8} T^{20} + 123 p^{9} T^{21} + 47 p^{10} T^{22} + 11 p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 - 7 T + 17 T^{2} + 7 T^{3} - 14 T^{5} - 1448 T^{6} + 5383 T^{7} + 13177 T^{8} - 74137 T^{9} + 167363 T^{10} - 476952 T^{11} + 2425844 T^{12} - 476952 p T^{13} + 167363 p^{2} T^{14} - 74137 p^{3} T^{15} + 13177 p^{4} T^{16} + 5383 p^{5} T^{17} - 1448 p^{6} T^{18} - 14 p^{7} T^{19} + 7 p^{9} T^{21} + 17 p^{10} T^{22} - 7 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 - 9 T - 9 T^{2} + 235 T^{3} + 440 T^{4} - 7052 T^{5} + 904 T^{6} + 93021 T^{7} + 805 p T^{8} - 1306603 T^{9} + 134215 p T^{10} + 3749024 T^{11} - 11467928 T^{12} + 3749024 p T^{13} + 134215 p^{3} T^{14} - 1306603 p^{3} T^{15} + 805 p^{5} T^{16} + 93021 p^{5} T^{17} + 904 p^{6} T^{18} - 7052 p^{7} T^{19} + 440 p^{8} T^{20} + 235 p^{9} T^{21} - 9 p^{10} T^{22} - 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 - 133 T^{2} + 522 p T^{4} - 22890 p T^{6} + 12426957 T^{8} - 303621745 T^{10} + 5811079504 T^{12} - 303621745 p^{2} T^{14} + 12426957 p^{4} T^{16} - 22890 p^{7} T^{18} + 522 p^{9} T^{20} - 133 p^{10} T^{22} + p^{12} T^{24} \) | |
| 19 | \( 1 + 7 T + 31 T^{2} + 189 T^{3} + 1426 T^{4} + 8162 T^{5} + 37516 T^{6} + 183911 T^{7} + 989579 T^{8} + 5075581 T^{9} + 23151601 T^{10} + 103810854 T^{11} + 453309164 T^{12} + 103810854 p T^{13} + 23151601 p^{2} T^{14} + 5075581 p^{3} T^{15} + 989579 p^{4} T^{16} + 183911 p^{5} T^{17} + 37516 p^{6} T^{18} + 8162 p^{7} T^{19} + 1426 p^{8} T^{20} + 189 p^{9} T^{21} + 31 p^{10} T^{22} + 7 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 + 5 T - 43 T^{2} - 17 p T^{3} - 308 T^{4} + 10696 T^{5} + 58590 T^{6} - 14345 T^{7} - 1422557 T^{8} - 5436135 T^{9} + 1966389 T^{10} + 79674994 T^{11} + 420307216 T^{12} + 79674994 p T^{13} + 1966389 p^{2} T^{14} - 5436135 p^{3} T^{15} - 1422557 p^{4} T^{16} - 14345 p^{5} T^{17} + 58590 p^{6} T^{18} + 10696 p^{7} T^{19} - 308 p^{8} T^{20} - 17 p^{10} T^{21} - 43 p^{10} T^{22} + 5 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 + 21 T + 269 T^{2} + 2261 T^{3} + 14784 T^{4} + 77658 T^{5} + 355230 T^{6} + 1212127 T^{7} - 208667 T^{8} - 52988005 T^{9} - 588286643 T^{10} - 4352370134 T^{11} - 26495076348 T^{12} - 4352370134 p T^{13} - 588286643 p^{2} T^{14} - 52988005 p^{3} T^{15} - 208667 p^{4} T^{16} + 1212127 p^{5} T^{17} + 355230 p^{6} T^{18} + 77658 p^{7} T^{19} + 14784 p^{8} T^{20} + 2261 p^{9} T^{21} + 269 p^{10} T^{22} + 21 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 - 7 T - 3 T^{2} + 833 T^{3} - 5462 T^{4} - 9800 T^{5} + 411870 T^{6} - 2163693 T^{7} - 6602285 T^{8} + 140542059 T^{9} - 562585377 T^{10} - 2805854072 T^{11} + 36132816832 T^{12} - 2805854072 p T^{13} - 562585377 p^{2} T^{14} + 140542059 p^{3} T^{15} - 6602285 p^{4} T^{16} - 2163693 p^{5} T^{17} + 411870 p^{6} T^{18} - 9800 p^{7} T^{19} - 5462 p^{8} T^{20} + 833 p^{9} T^{21} - 3 p^{10} T^{22} - 7 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 - 393 T^{2} + 73710 T^{4} - 8727326 T^{6} + 726625577 T^{8} - 44820888217 T^{10} + 51228753680 p T^{12} - 44820888217 p^{2} T^{14} + 726625577 p^{4} T^{16} - 8727326 p^{6} T^{18} + 73710 p^{8} T^{20} - 393 p^{10} T^{22} + p^{12} T^{24} \) | |
| 43 | \( 1 - 7 T + 139 T^{2} - 525 T^{3} + 4746 T^{4} + 19698 T^{5} - 5298 p T^{6} + 3326155 T^{7} - 15690613 T^{8} + 60933523 T^{9} + 454409879 T^{10} - 6181766668 T^{11} + 56674975708 T^{12} - 6181766668 p T^{13} + 454409879 p^{2} T^{14} + 60933523 p^{3} T^{15} - 15690613 p^{4} T^{16} + 3326155 p^{5} T^{17} - 5298 p^{7} T^{18} + 19698 p^{7} T^{19} + 4746 p^{8} T^{20} - 525 p^{9} T^{21} + 139 p^{10} T^{22} - 7 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 + 7 T - 9 T^{2} - 497 T^{3} - 5250 T^{4} - 33334 T^{5} - 12592 T^{6} + 1824669 T^{7} + 19098315 T^{8} + 97990109 T^{9} + 127809129 T^{10} - 4616139010 T^{11} - 56042219764 T^{12} - 4616139010 p T^{13} + 127809129 p^{2} T^{14} + 97990109 p^{3} T^{15} + 19098315 p^{4} T^{16} + 1824669 p^{5} T^{17} - 12592 p^{6} T^{18} - 33334 p^{7} T^{19} - 5250 p^{8} T^{20} - 497 p^{9} T^{21} - 9 p^{10} T^{22} + 7 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 + 10 T + 25 T^{2} - 278 T^{3} + 2512 T^{4} + 20294 T^{5} - 44832 T^{6} - 1725298 T^{7} + 3740845 T^{8} + 15939776 T^{9} - 71501305 T^{10} - 3189845416 T^{11} + 681878948 T^{12} - 3189845416 p T^{13} - 71501305 p^{2} T^{14} + 15939776 p^{3} T^{15} + 3740845 p^{4} T^{16} - 1725298 p^{5} T^{17} - 44832 p^{6} T^{18} + 20294 p^{7} T^{19} + 2512 p^{8} T^{20} - 278 p^{9} T^{21} + 25 p^{10} T^{22} + 10 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( ( 1 - 22 T + 446 T^{2} - 6050 T^{3} + 72311 T^{4} - 688228 T^{5} + 5840260 T^{6} - 688228 p T^{7} + 72311 p^{2} T^{8} - 6050 p^{3} T^{9} + 446 p^{4} T^{10} - 22 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 61 | \( 1 + 7 T + 159 T^{2} + 105 T^{3} + 5012 T^{4} - 64120 T^{5} + 241116 T^{6} - 1456693 T^{7} + 53389705 T^{8} + 83034021 T^{9} + 2548004823 T^{10} - 10608027248 T^{11} + 41335695248 T^{12} - 10608027248 p T^{13} + 2548004823 p^{2} T^{14} + 83034021 p^{3} T^{15} + 53389705 p^{4} T^{16} - 1456693 p^{5} T^{17} + 241116 p^{6} T^{18} - 64120 p^{7} T^{19} + 5012 p^{8} T^{20} + 105 p^{9} T^{21} + 159 p^{10} T^{22} + 7 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 + 37 T + 513 T^{2} + 3025 T^{3} - 450 T^{4} - 212780 T^{5} - 3203838 T^{6} - 28238209 T^{7} - 156569483 T^{8} - 254147545 T^{9} + 8082101273 T^{10} + 110845067132 T^{11} + 930583608704 T^{12} + 110845067132 p T^{13} + 8082101273 p^{2} T^{14} - 254147545 p^{3} T^{15} - 156569483 p^{4} T^{16} - 28238209 p^{5} T^{17} - 3203838 p^{6} T^{18} - 212780 p^{7} T^{19} - 450 p^{8} T^{20} + 3025 p^{9} T^{21} + 513 p^{10} T^{22} + 37 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 + 21 T + 173 T^{2} + 721 T^{3} - 4778 T^{4} - 187446 T^{5} - 2306210 T^{6} - 3297 p^{2} T^{7} - 82259473 T^{8} + 61660641 T^{9} + 7528298205 T^{10} + 1293193300 p T^{11} + 11332940220 p T^{12} + 1293193300 p^{2} T^{13} + 7528298205 p^{2} T^{14} + 61660641 p^{3} T^{15} - 82259473 p^{4} T^{16} - 3297 p^{7} T^{17} - 2306210 p^{6} T^{18} - 187446 p^{7} T^{19} - 4778 p^{8} T^{20} + 721 p^{9} T^{21} + 173 p^{10} T^{22} + 21 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 - 14 T + 223 T^{2} - 1638 T^{3} + 14902 T^{4} - 24080 T^{5} - 61436 T^{6} + 7903616 T^{7} - 50738367 T^{8} + 656032846 T^{9} - 446390355 T^{10} + 13646495926 T^{11} + 186938570672 T^{12} + 13646495926 p T^{13} - 446390355 p^{2} T^{14} + 656032846 p^{3} T^{15} - 50738367 p^{4} T^{16} + 7903616 p^{5} T^{17} - 61436 p^{6} T^{18} - 24080 p^{7} T^{19} + 14902 p^{8} T^{20} - 1638 p^{9} T^{21} + 223 p^{10} T^{22} - 14 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 - 49 T + 1401 T^{2} - 29659 T^{3} + 510220 T^{4} - 7424550 T^{5} + 93952026 T^{6} - 1051207157 T^{7} + 10574913669 T^{8} - 97442137415 T^{9} + 846233526101 T^{10} - 7200350040954 T^{11} + 62681377749836 T^{12} - 7200350040954 p T^{13} + 846233526101 p^{2} T^{14} - 97442137415 p^{3} T^{15} + 10574913669 p^{4} T^{16} - 1051207157 p^{5} T^{17} + 93952026 p^{6} T^{18} - 7424550 p^{7} T^{19} + 510220 p^{8} T^{20} - 29659 p^{9} T^{21} + 1401 p^{10} T^{22} - 49 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 - 5 T + 19 T^{2} - 291 T^{3} + 6964 T^{4} - 112648 T^{5} + 680674 T^{6} - 2772355 T^{7} + 73849055 T^{8} - 851751547 T^{9} + 7543184631 T^{10} - 61831617966 T^{11} + 242863289344 T^{12} - 61831617966 p T^{13} + 7543184631 p^{2} T^{14} - 851751547 p^{3} T^{15} + 73849055 p^{4} T^{16} - 2772355 p^{5} T^{17} + 680674 p^{6} T^{18} - 112648 p^{7} T^{19} + 6964 p^{8} T^{20} - 291 p^{9} T^{21} + 19 p^{10} T^{22} - 5 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 - 7 T + 319 T^{2} - 2107 T^{3} + 41592 T^{4} - 268912 T^{5} + 2519080 T^{6} - 17316453 T^{7} - 32462455 T^{8} + 662290951 T^{9} - 28551782921 T^{10} + 290592922368 T^{11} - 3660702381280 T^{12} + 290592922368 p T^{13} - 28551782921 p^{2} T^{14} + 662290951 p^{3} T^{15} - 32462455 p^{4} T^{16} - 17316453 p^{5} T^{17} + 2519080 p^{6} T^{18} - 268912 p^{7} T^{19} + 41592 p^{8} T^{20} - 2107 p^{9} T^{21} + 319 p^{10} T^{22} - 7 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 - 14 T + 441 T^{2} - 5068 T^{3} + 83194 T^{4} - 784252 T^{5} + 9216214 T^{6} - 758786 p T^{7} + 720487153 T^{8} - 5423464648 T^{9} + 47440764385 T^{10} - 407534937824 T^{11} + 3645124614984 T^{12} - 407534937824 p T^{13} + 47440764385 p^{2} T^{14} - 5423464648 p^{3} T^{15} + 720487153 p^{4} T^{16} - 758786 p^{6} T^{17} + 9216214 p^{6} T^{18} - 784252 p^{7} T^{19} + 83194 p^{8} T^{20} - 5068 p^{9} T^{21} + 441 p^{10} T^{22} - 14 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−7.08283534664582326321460632384, −6.83506554588557243892079035888, −6.70990106790263234729638205098, −6.70726854227879352784738010540, −6.50583116542003016405616963010, −6.29086570761762050849236864710, −6.23380234083278213348474027600, −6.20632046541944219261412287523, −5.95756030164411865972027242283, −5.82135922482034502991718014869, −5.73960901976586531278033097929, −5.62874404433070015118395725525, −5.40093903660652915883022383970, −5.08858261477480530779716134548, −4.99381921463705267630500825165, −4.79104683189176889556865831916, −4.08029133751381065579970008725, −4.04433579375012822567938171473, −4.03063395235420135060249814795, −3.95733245259119593742700930144, −3.64588921002318162183508342745, −3.33608310731647762704031487964, −3.07956168117191385141463103997, −2.33858877236063270087836019945, −1.44050764825646200035269621137, 1.44050764825646200035269621137, 2.33858877236063270087836019945, 3.07956168117191385141463103997, 3.33608310731647762704031487964, 3.64588921002318162183508342745, 3.95733245259119593742700930144, 4.03063395235420135060249814795, 4.04433579375012822567938171473, 4.08029133751381065579970008725, 4.79104683189176889556865831916, 4.99381921463705267630500825165, 5.08858261477480530779716134548, 5.40093903660652915883022383970, 5.62874404433070015118395725525, 5.73960901976586531278033097929, 5.82135922482034502991718014869, 5.95756030164411865972027242283, 6.20632046541944219261412287523, 6.23380234083278213348474027600, 6.29086570761762050849236864710, 6.50583116542003016405616963010, 6.70726854227879352784738010540, 6.70990106790263234729638205098, 6.83506554588557243892079035888, 7.08283534664582326321460632384
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.