Dirichlet series
| L(s) = 1 | − 2·2-s + 12·3-s − 5·4-s + 12·5-s − 24·6-s + 14·8-s + 78·9-s − 24·10-s + 10·11-s − 60·12-s + 15·13-s + 144·15-s + 7·16-s + 8·17-s − 156·18-s + 2·19-s − 60·20-s − 20·22-s + 5·23-s + 168·24-s + 52·25-s − 30·26-s + 364·27-s + 20·29-s − 288·30-s + 10·31-s − 43·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 6.92·3-s − 5/2·4-s + 5.36·5-s − 9.79·6-s + 4.94·8-s + 26·9-s − 7.58·10-s + 3.01·11-s − 17.3·12-s + 4.16·13-s + 37.1·15-s + 7/4·16-s + 1.94·17-s − 36.7·18-s + 0.458·19-s − 13.4·20-s − 4.26·22-s + 1.04·23-s + 34.2·24-s + 52/5·25-s − 5.88·26-s + 70.0·27-s + 3.71·29-s − 52.5·30-s + 1.79·31-s − 7.60·32-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{24} \cdot 41^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{24} \cdot 41^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(3^{12} \cdot 7^{24} \cdot 41^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.54361\times 10^{20}\) |
| Root analytic conductor: | \(6.93727\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 3^{12} \cdot 7^{24} \cdot 41^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(27176.48159\) |
| \(L(\frac12)\) | \(\approx\) | \(27176.48159\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 - T )^{12} \) |
| 7 | \( 1 \) | |
| 41 | \( ( 1 + T )^{12} \) | |
| good | 2 | \( 1 + p T + 9 T^{2} + 7 p T^{3} + 19 p T^{4} + 49 T^{5} + 13 p^{3} T^{6} + p^{7} T^{7} + 127 p T^{8} + 347 T^{9} + 5 p^{7} T^{10} + 901 T^{11} + 721 p T^{12} + 901 p T^{13} + 5 p^{9} T^{14} + 347 p^{3} T^{15} + 127 p^{5} T^{16} + p^{12} T^{17} + 13 p^{9} T^{18} + 49 p^{7} T^{19} + 19 p^{9} T^{20} + 7 p^{10} T^{21} + 9 p^{10} T^{22} + p^{12} T^{23} + p^{12} T^{24} \) |
| 5 | \( 1 - 12 T + 92 T^{2} - 103 p T^{3} + 473 p T^{4} - 9276 T^{5} + 32308 T^{6} - 101928 T^{7} + 296764 T^{8} - 805406 T^{9} + 2057529 T^{10} - 4969294 T^{11} + 11401404 T^{12} - 4969294 p T^{13} + 2057529 p^{2} T^{14} - 805406 p^{3} T^{15} + 296764 p^{4} T^{16} - 101928 p^{5} T^{17} + 32308 p^{6} T^{18} - 9276 p^{7} T^{19} + 473 p^{9} T^{20} - 103 p^{10} T^{21} + 92 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 - 10 T + 122 T^{2} - 859 T^{3} + 6293 T^{4} - 35149 T^{5} + 196101 T^{6} - 919472 T^{7} + 4253857 T^{8} - 17248739 T^{9} + 68808934 T^{10} - 244979028 T^{11} + 857874634 T^{12} - 244979028 p T^{13} + 68808934 p^{2} T^{14} - 17248739 p^{3} T^{15} + 4253857 p^{4} T^{16} - 919472 p^{5} T^{17} + 196101 p^{6} T^{18} - 35149 p^{7} T^{19} + 6293 p^{8} T^{20} - 859 p^{9} T^{21} + 122 p^{10} T^{22} - 10 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 - 15 T + 180 T^{2} - 1602 T^{3} + 12336 T^{4} - 82204 T^{5} + 493723 T^{6} - 2684087 T^{7} + 13434542 T^{8} - 62013611 T^{9} + 266360236 T^{10} - 1064138421 T^{11} + 3973246191 T^{12} - 1064138421 p T^{13} + 266360236 p^{2} T^{14} - 62013611 p^{3} T^{15} + 13434542 p^{4} T^{16} - 2684087 p^{5} T^{17} + 493723 p^{6} T^{18} - 82204 p^{7} T^{19} + 12336 p^{8} T^{20} - 1602 p^{9} T^{21} + 180 p^{10} T^{22} - 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 - 8 T + 143 T^{2} - 905 T^{3} + 9595 T^{4} - 51295 T^{5} + 414138 T^{6} - 1935224 T^{7} + 12986717 T^{8} - 53904784 T^{9} + 313122480 T^{10} - 1161451435 T^{11} + 5964061488 T^{12} - 1161451435 p T^{13} + 313122480 p^{2} T^{14} - 53904784 p^{3} T^{15} + 12986717 p^{4} T^{16} - 1935224 p^{5} T^{17} + 414138 p^{6} T^{18} - 51295 p^{7} T^{19} + 9595 p^{8} T^{20} - 905 p^{9} T^{21} + 143 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 - 2 T + 134 T^{2} - 246 T^{3} + 9080 T^{4} - 14987 T^{5} + 410832 T^{6} - 610501 T^{7} + 13838781 T^{8} - 18615040 T^{9} + 365259703 T^{10} - 444709835 T^{11} + 7727373535 T^{12} - 444709835 p T^{13} + 365259703 p^{2} T^{14} - 18615040 p^{3} T^{15} + 13838781 p^{4} T^{16} - 610501 p^{5} T^{17} + 410832 p^{6} T^{18} - 14987 p^{7} T^{19} + 9080 p^{8} T^{20} - 246 p^{9} T^{21} + 134 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 - 5 T + 150 T^{2} - 756 T^{3} + 457 p T^{4} - 53944 T^{5} + 485040 T^{6} - 2486031 T^{7} + 17672611 T^{8} - 85504218 T^{9} + 545081844 T^{10} - 2368518321 T^{11} + 13931123614 T^{12} - 2368518321 p T^{13} + 545081844 p^{2} T^{14} - 85504218 p^{3} T^{15} + 17672611 p^{4} T^{16} - 2486031 p^{5} T^{17} + 485040 p^{6} T^{18} - 53944 p^{7} T^{19} + 457 p^{9} T^{20} - 756 p^{9} T^{21} + 150 p^{10} T^{22} - 5 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 - 20 T + 447 T^{2} - 5790 T^{3} + 75707 T^{4} - 734382 T^{5} + 7075290 T^{6} - 55182589 T^{7} + 428753960 T^{8} - 2813471809 T^{9} + 18559828017 T^{10} - 105761092457 T^{11} + 610479409448 T^{12} - 105761092457 p T^{13} + 18559828017 p^{2} T^{14} - 2813471809 p^{3} T^{15} + 428753960 p^{4} T^{16} - 55182589 p^{5} T^{17} + 7075290 p^{6} T^{18} - 734382 p^{7} T^{19} + 75707 p^{8} T^{20} - 5790 p^{9} T^{21} + 447 p^{10} T^{22} - 20 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 - 10 T + 199 T^{2} - 1858 T^{3} + 21309 T^{4} - 174066 T^{5} + 1545369 T^{6} - 11136866 T^{7} + 83954164 T^{8} - 540471396 T^{9} + 3596251098 T^{10} - 20812542847 T^{11} + 123921349927 T^{12} - 20812542847 p T^{13} + 3596251098 p^{2} T^{14} - 540471396 p^{3} T^{15} + 83954164 p^{4} T^{16} - 11136866 p^{5} T^{17} + 1545369 p^{6} T^{18} - 174066 p^{7} T^{19} + 21309 p^{8} T^{20} - 1858 p^{9} T^{21} + 199 p^{10} T^{22} - 10 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 + 17 T + 454 T^{2} + 5943 T^{3} + 91024 T^{4} + 972522 T^{5} + 10939811 T^{6} + 98437595 T^{7} + 888838768 T^{8} + 6849829509 T^{9} + 51754214452 T^{10} + 344078250757 T^{11} + 2221085359833 T^{12} + 344078250757 p T^{13} + 51754214452 p^{2} T^{14} + 6849829509 p^{3} T^{15} + 888838768 p^{4} T^{16} + 98437595 p^{5} T^{17} + 10939811 p^{6} T^{18} + 972522 p^{7} T^{19} + 91024 p^{8} T^{20} + 5943 p^{9} T^{21} + 454 p^{10} T^{22} + 17 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 - 12 T + 491 T^{2} - 5340 T^{3} + 112078 T^{4} - 1091529 T^{5} + 15680202 T^{6} - 135428034 T^{7} + 1493772821 T^{8} - 11342317651 T^{9} + 101896975115 T^{10} - 673713714338 T^{11} + 5097914416640 T^{12} - 673713714338 p T^{13} + 101896975115 p^{2} T^{14} - 11342317651 p^{3} T^{15} + 1493772821 p^{4} T^{16} - 135428034 p^{5} T^{17} + 15680202 p^{6} T^{18} - 1091529 p^{7} T^{19} + 112078 p^{8} T^{20} - 5340 p^{9} T^{21} + 491 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 - 34 T + 881 T^{2} - 15591 T^{3} + 231671 T^{4} - 2754578 T^{5} + 28411476 T^{6} - 243863530 T^{7} + 1835802451 T^{8} - 11559516963 T^{9} + 65909573965 T^{10} - 339401469527 T^{11} + 2106040442704 T^{12} - 339401469527 p T^{13} + 65909573965 p^{2} T^{14} - 11559516963 p^{3} T^{15} + 1835802451 p^{4} T^{16} - 243863530 p^{5} T^{17} + 28411476 p^{6} T^{18} - 2754578 p^{7} T^{19} + 231671 p^{8} T^{20} - 15591 p^{9} T^{21} + 881 p^{10} T^{22} - 34 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 - 6 T + 146 T^{2} - 1152 T^{3} + 362 p T^{4} - 98075 T^{5} + 1430136 T^{6} - 7347067 T^{7} + 86373879 T^{8} - 295437005 T^{9} + 4581724908 T^{10} - 260195426 p T^{11} + 213176736460 T^{12} - 260195426 p^{2} T^{13} + 4581724908 p^{2} T^{14} - 295437005 p^{3} T^{15} + 86373879 p^{4} T^{16} - 7347067 p^{5} T^{17} + 1430136 p^{6} T^{18} - 98075 p^{7} T^{19} + 362 p^{9} T^{20} - 1152 p^{9} T^{21} + 146 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 - 27 T + 589 T^{2} - 8778 T^{3} + 119080 T^{4} - 1352787 T^{5} + 14660959 T^{6} - 143669444 T^{7} + 1372588760 T^{8} - 12257086778 T^{9} + 106631600267 T^{10} - 872626167317 T^{11} + 6909808756292 T^{12} - 872626167317 p T^{13} + 106631600267 p^{2} T^{14} - 12257086778 p^{3} T^{15} + 1372588760 p^{4} T^{16} - 143669444 p^{5} T^{17} + 14660959 p^{6} T^{18} - 1352787 p^{7} T^{19} + 119080 p^{8} T^{20} - 8778 p^{9} T^{21} + 589 p^{10} T^{22} - 27 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 - 22 T + 650 T^{2} - 10150 T^{3} + 177132 T^{4} - 2194671 T^{5} + 28972148 T^{6} - 303073496 T^{7} + 3327301613 T^{8} - 30439272193 T^{9} + 290350986372 T^{10} - 2358415248533 T^{11} + 19903180424514 T^{12} - 2358415248533 p T^{13} + 290350986372 p^{2} T^{14} - 30439272193 p^{3} T^{15} + 3327301613 p^{4} T^{16} - 303073496 p^{5} T^{17} + 28972148 p^{6} T^{18} - 2194671 p^{7} T^{19} + 177132 p^{8} T^{20} - 10150 p^{9} T^{21} + 650 p^{10} T^{22} - 22 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 + 26 T + 495 T^{2} + 6782 T^{3} + 88979 T^{4} + 1013776 T^{5} + 11332027 T^{6} + 112686208 T^{7} + 1114572440 T^{8} + 10111260790 T^{9} + 91986623724 T^{10} + 771262209217 T^{11} + 6539609167373 T^{12} + 771262209217 p T^{13} + 91986623724 p^{2} T^{14} + 10111260790 p^{3} T^{15} + 1114572440 p^{4} T^{16} + 112686208 p^{5} T^{17} + 11332027 p^{6} T^{18} + 1013776 p^{7} T^{19} + 88979 p^{8} T^{20} + 6782 p^{9} T^{21} + 495 p^{10} T^{22} + 26 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 - 50 T + 1628 T^{2} - 38427 T^{3} + 748398 T^{4} - 12346666 T^{5} + 180064873 T^{6} - 2345231472 T^{7} + 27854213111 T^{8} - 302956955874 T^{9} + 3054383087961 T^{10} - 28558706193300 T^{11} + 249382706903706 T^{12} - 28558706193300 p T^{13} + 3054383087961 p^{2} T^{14} - 302956955874 p^{3} T^{15} + 27854213111 p^{4} T^{16} - 2345231472 p^{5} T^{17} + 180064873 p^{6} T^{18} - 12346666 p^{7} T^{19} + 748398 p^{8} T^{20} - 38427 p^{9} T^{21} + 1628 p^{10} T^{22} - 50 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 - 21 T + 378 T^{2} - 3933 T^{3} + 51242 T^{4} - 488895 T^{5} + 6036139 T^{6} - 50218882 T^{7} + 574462735 T^{8} - 4702122347 T^{9} + 52877947565 T^{10} - 397687085357 T^{11} + 4060761766823 T^{12} - 397687085357 p T^{13} + 52877947565 p^{2} T^{14} - 4702122347 p^{3} T^{15} + 574462735 p^{4} T^{16} - 50218882 p^{5} T^{17} + 6036139 p^{6} T^{18} - 488895 p^{7} T^{19} + 51242 p^{8} T^{20} - 3933 p^{9} T^{21} + 378 p^{10} T^{22} - 21 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 + 10 T + 507 T^{2} + 6230 T^{3} + 138295 T^{4} + 1781737 T^{5} + 26727811 T^{6} + 323237069 T^{7} + 3929073146 T^{8} + 42288957847 T^{9} + 448931801426 T^{10} + 4239370736409 T^{11} + 40105045005161 T^{12} + 4239370736409 p T^{13} + 448931801426 p^{2} T^{14} + 42288957847 p^{3} T^{15} + 3929073146 p^{4} T^{16} + 323237069 p^{5} T^{17} + 26727811 p^{6} T^{18} + 1781737 p^{7} T^{19} + 138295 p^{8} T^{20} + 6230 p^{9} T^{21} + 507 p^{10} T^{22} + 10 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 - 8 T + 494 T^{2} - 3684 T^{3} + 125170 T^{4} - 945165 T^{5} + 21866508 T^{6} - 169564051 T^{7} + 2932914155 T^{8} - 22867946991 T^{9} + 318753337090 T^{10} - 2402739841042 T^{11} + 28871280552130 T^{12} - 2402739841042 p T^{13} + 318753337090 p^{2} T^{14} - 22867946991 p^{3} T^{15} + 2932914155 p^{4} T^{16} - 169564051 p^{5} T^{17} + 21866508 p^{6} T^{18} - 945165 p^{7} T^{19} + 125170 p^{8} T^{20} - 3684 p^{9} T^{21} + 494 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 - 11 T + 662 T^{2} - 6250 T^{3} + 2353 p T^{4} - 1700506 T^{5} + 42117397 T^{6} - 296145672 T^{7} + 6121801761 T^{8} - 37792220344 T^{9} + 700838829954 T^{10} - 3899665613862 T^{11} + 67271529136428 T^{12} - 3899665613862 p T^{13} + 700838829954 p^{2} T^{14} - 37792220344 p^{3} T^{15} + 6121801761 p^{4} T^{16} - 296145672 p^{5} T^{17} + 42117397 p^{6} T^{18} - 1700506 p^{7} T^{19} + 2353 p^{9} T^{20} - 6250 p^{9} T^{21} + 662 p^{10} T^{22} - 11 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 - 32 T + 1150 T^{2} - 23853 T^{3} + 512829 T^{4} - 8204160 T^{5} + 134546662 T^{6} - 1800915850 T^{7} + 24653184852 T^{8} - 287298613908 T^{9} + 3418851179051 T^{10} - 35315386861568 T^{11} + 372231681881580 T^{12} - 35315386861568 p T^{13} + 3418851179051 p^{2} T^{14} - 287298613908 p^{3} T^{15} + 24653184852 p^{4} T^{16} - 1800915850 p^{5} T^{17} + 134546662 p^{6} T^{18} - 8204160 p^{7} T^{19} + 512829 p^{8} T^{20} - 23853 p^{9} T^{21} + 1150 p^{10} T^{22} - 32 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−2.33990906099518597302816217572, −2.30256571186632807949546131279, −2.17396052842879330201523088166, −2.14473050665652202880723456304, −2.10177746968632723446912062423, −1.96462015304087928909311955794, −1.81885261424978526673163822308, −1.81343971304922799863466891362, −1.76391377975268501923766831510, −1.62922203806142096786816825473, −1.55636247132475207625626571048, −1.53314763618229745064732484148, −1.52884156475195510082116115697, −1.45364649243097288173264992650, −1.41081539938563807507537258002, −1.02912110860940543089711797181, −0.972837937599556189107406692433, −0.957787657259516074307801453580, −0.933789212673705436567896640513, −0.917756246479829720450446716704, −0.71530261441748584010920948808, −0.68366170361068228039093143685, −0.60290985493430040258104422520, −0.55192210315641414423167203250, −0.52987945796703708076159945332, 0.52987945796703708076159945332, 0.55192210315641414423167203250, 0.60290985493430040258104422520, 0.68366170361068228039093143685, 0.71530261441748584010920948808, 0.917756246479829720450446716704, 0.933789212673705436567896640513, 0.957787657259516074307801453580, 0.972837937599556189107406692433, 1.02912110860940543089711797181, 1.41081539938563807507537258002, 1.45364649243097288173264992650, 1.52884156475195510082116115697, 1.53314763618229745064732484148, 1.55636247132475207625626571048, 1.62922203806142096786816825473, 1.76391377975268501923766831510, 1.81343971304922799863466891362, 1.81885261424978526673163822308, 1.96462015304087928909311955794, 2.10177746968632723446912062423, 2.14473050665652202880723456304, 2.17396052842879330201523088166, 2.30256571186632807949546131279, 2.33990906099518597302816217572
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.