Dirichlet series
| L(s) = 1 | + 12·5-s − 4·7-s − 6·9-s − 4·11-s + 5·16-s − 8·17-s + 72·25-s − 8·29-s + 24·31-s + 4·32-s − 48·35-s − 4·37-s − 20·41-s − 72·45-s + 32·47-s + 8·49-s − 16·53-s − 48·55-s + 8·59-s + 24·63-s − 32·67-s − 12·71-s − 32·73-s + 16·77-s + 24·79-s + 60·80-s + 21·81-s + ⋯ |
| L(s) = 1 | + 5.36·5-s − 1.51·7-s − 2·9-s − 1.20·11-s + 5/4·16-s − 1.94·17-s + 72/5·25-s − 1.48·29-s + 4.31·31-s + 0.707·32-s − 8.11·35-s − 0.657·37-s − 3.12·41-s − 10.7·45-s + 4.66·47-s + 8/7·49-s − 2.19·53-s − 6.47·55-s + 1.04·59-s + 3.02·63-s − 3.90·67-s − 1.42·71-s − 3.74·73-s + 1.82·77-s + 2.70·79-s + 6.70·80-s + 7/3·81-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
| \(L(1)\) | \(\approx\) | \(1.186881932\) |
| \(L(\frac12)\) | \(\approx\) | \(1.186881932\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 + T^{2} )^{6} \) |
| 7 | \( 1 + 4 T + 8 T^{2} + 4 T^{3} + 19 T^{4} - 128 T^{5} - 528 T^{6} - 128 p T^{7} + 19 p^{2} T^{8} + 4 p^{3} T^{9} + 8 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \) | |
| 13 | \( 1 - 10 T^{2} + 159 T^{4} + 288 T^{5} - 3020 T^{6} + 288 p T^{7} + 159 p^{2} T^{8} - 10 p^{4} T^{10} + p^{6} T^{12} \) | |
| good | 2 | \( 1 - 5 T^{4} - p^{2} T^{5} + 3 p^{2} T^{7} + 31 T^{8} + 5 p^{2} T^{9} + p^{3} T^{10} - 7 p^{2} T^{11} - 167 T^{12} - 7 p^{3} T^{13} + p^{5} T^{14} + 5 p^{5} T^{15} + 31 p^{4} T^{16} + 3 p^{7} T^{17} - p^{9} T^{19} - 5 p^{8} T^{20} + p^{12} T^{24} \) |
| 5 | \( 1 - 12 T + 72 T^{2} - 12 p^{2} T^{3} + 206 p T^{4} - 644 p T^{5} + 1896 p T^{6} - 26148 T^{7} + 539 p^{3} T^{8} - 6592 p^{2} T^{9} + 15632 p^{2} T^{10} - 181232 p T^{11} + 2051716 T^{12} - 181232 p^{2} T^{13} + 15632 p^{4} T^{14} - 6592 p^{5} T^{15} + 539 p^{7} T^{16} - 26148 p^{5} T^{17} + 1896 p^{7} T^{18} - 644 p^{8} T^{19} + 206 p^{9} T^{20} - 12 p^{11} T^{21} + 72 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + 194 T^{4} + 268 T^{5} + 488 T^{6} + 236 p T^{7} - 10193 T^{8} - 76496 T^{9} - 168976 T^{10} - 96336 p T^{11} - 53908 p^{2} T^{12} - 96336 p^{2} T^{13} - 168976 p^{2} T^{14} - 76496 p^{3} T^{15} - 10193 p^{4} T^{16} + 236 p^{6} T^{17} + 488 p^{6} T^{18} + 268 p^{7} T^{19} + 194 p^{8} T^{20} + 4 p^{10} T^{21} + 8 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( ( 1 + 4 T + 4 p T^{2} + 12 p T^{3} + 2259 T^{4} + 5392 T^{5} + 46632 T^{6} + 5392 p T^{7} + 2259 p^{2} T^{8} + 12 p^{4} T^{9} + 4 p^{5} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 19 | \( 1 - 64 T^{3} - 74 T^{4} + 2176 T^{5} + 2048 T^{6} + 41664 T^{7} - 171137 T^{8} - 621888 T^{9} - 147456 T^{10} - 1444928 T^{11} + 107862004 T^{12} - 1444928 p T^{13} - 147456 p^{2} T^{14} - 621888 p^{3} T^{15} - 171137 p^{4} T^{16} + 41664 p^{5} T^{17} + 2048 p^{6} T^{18} + 2176 p^{7} T^{19} - 74 p^{8} T^{20} - 64 p^{9} T^{21} + p^{12} T^{24} \) | |
| 23 | \( 1 - 8 p T^{2} + 16630 T^{4} - 979576 T^{6} + 41951151 T^{8} - 1377491536 T^{10} + 35582296884 T^{12} - 1377491536 p^{2} T^{14} + 41951151 p^{4} T^{16} - 979576 p^{6} T^{18} + 16630 p^{8} T^{20} - 8 p^{11} T^{22} + p^{12} T^{24} \) | |
| 29 | \( ( 1 + 4 T + 102 T^{2} + 324 T^{3} + 5255 T^{4} + 13320 T^{5} + 181652 T^{6} + 13320 p T^{7} + 5255 p^{2} T^{8} + 324 p^{3} T^{9} + 102 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 31 | \( 1 - 24 T + 288 T^{2} - 2664 T^{3} + 23366 T^{4} - 191480 T^{5} + 1414560 T^{6} - 9833864 T^{7} + 65863567 T^{8} - 416856560 T^{9} + 2512950080 T^{10} - 14801213200 T^{11} + 84424185236 T^{12} - 14801213200 p T^{13} + 2512950080 p^{2} T^{14} - 416856560 p^{3} T^{15} + 65863567 p^{4} T^{16} - 9833864 p^{5} T^{17} + 1414560 p^{6} T^{18} - 191480 p^{7} T^{19} + 23366 p^{8} T^{20} - 2664 p^{9} T^{21} + 288 p^{10} T^{22} - 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 + 4 T + 8 T^{2} - 4 p T^{3} - 5734 T^{4} - 18804 T^{5} - 18392 T^{6} + 642628 T^{7} + 14980447 T^{8} + 38852008 T^{9} + 11790992 T^{10} - 1372315080 T^{11} - 24637465492 T^{12} - 1372315080 p T^{13} + 11790992 p^{2} T^{14} + 38852008 p^{3} T^{15} + 14980447 p^{4} T^{16} + 642628 p^{5} T^{17} - 18392 p^{6} T^{18} - 18804 p^{7} T^{19} - 5734 p^{8} T^{20} - 4 p^{10} T^{21} + 8 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 + 20 T + 200 T^{2} + 1396 T^{3} + 6486 T^{4} + 21068 T^{5} + 98568 T^{6} + 657020 T^{7} + 3400383 T^{8} + 18798928 T^{9} + 195720144 T^{10} + 2403191168 T^{11} + 20018748580 T^{12} + 2403191168 p T^{13} + 195720144 p^{2} T^{14} + 18798928 p^{3} T^{15} + 3400383 p^{4} T^{16} + 657020 p^{5} T^{17} + 98568 p^{6} T^{18} + 21068 p^{7} T^{19} + 6486 p^{8} T^{20} + 1396 p^{9} T^{21} + 200 p^{10} T^{22} + 20 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 - 244 T^{2} + 30610 T^{4} - 2638596 T^{6} + 175601343 T^{8} - 9605919592 T^{10} + 446183179324 T^{12} - 9605919592 p^{2} T^{14} + 175601343 p^{4} T^{16} - 2638596 p^{6} T^{18} + 30610 p^{8} T^{20} - 244 p^{10} T^{22} + p^{12} T^{24} \) | |
| 47 | \( 1 - 32 T + 512 T^{2} - 5560 T^{3} + 44382 T^{4} - 263568 T^{5} + 1167392 T^{6} - 3425432 T^{7} + 6738543 T^{8} - 130913688 T^{9} + 2707359488 T^{10} - 33004404056 T^{11} + 272479402868 T^{12} - 33004404056 p T^{13} + 2707359488 p^{2} T^{14} - 130913688 p^{3} T^{15} + 6738543 p^{4} T^{16} - 3425432 p^{5} T^{17} + 1167392 p^{6} T^{18} - 263568 p^{7} T^{19} + 44382 p^{8} T^{20} - 5560 p^{9} T^{21} + 512 p^{10} T^{22} - 32 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( ( 1 + 8 T + 298 T^{2} + 1800 T^{3} + 37015 T^{4} + 173072 T^{5} + 2546988 T^{6} + 173072 p T^{7} + 37015 p^{2} T^{8} + 1800 p^{3} T^{9} + 298 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 59 | \( 1 - 8 T + 32 T^{2} + 48 T^{3} - 54 p T^{4} + 6040 T^{5} + 54784 T^{6} - 409248 T^{7} - 7406785 T^{8} + 79831816 T^{9} - 228498688 T^{10} - 21907048 p T^{11} + 14248180 p^{2} T^{12} - 21907048 p^{2} T^{13} - 228498688 p^{2} T^{14} + 79831816 p^{3} T^{15} - 7406785 p^{4} T^{16} - 409248 p^{5} T^{17} + 54784 p^{6} T^{18} + 6040 p^{7} T^{19} - 54 p^{9} T^{20} + 48 p^{9} T^{21} + 32 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 - 236 T^{2} + 28482 T^{4} - 2427132 T^{6} + 186697823 T^{8} - 13906047096 T^{10} + 927136000732 T^{12} - 13906047096 p^{2} T^{14} + 186697823 p^{4} T^{16} - 2427132 p^{6} T^{18} + 28482 p^{8} T^{20} - 236 p^{10} T^{22} + p^{12} T^{24} \) | |
| 67 | \( 1 + 32 T + 512 T^{2} + 6240 T^{3} + 75670 T^{4} + 928160 T^{5} + 10426880 T^{6} + 108246240 T^{7} + 1063815999 T^{8} + 9884709184 T^{9} + 19788800 p^{2} T^{10} + 11789889856 p T^{11} + 6724545443124 T^{12} + 11789889856 p^{2} T^{13} + 19788800 p^{4} T^{14} + 9884709184 p^{3} T^{15} + 1063815999 p^{4} T^{16} + 108246240 p^{5} T^{17} + 10426880 p^{6} T^{18} + 928160 p^{7} T^{19} + 75670 p^{8} T^{20} + 6240 p^{9} T^{21} + 512 p^{10} T^{22} + 32 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 + 12 T + 72 T^{2} + 1156 T^{3} + 15698 T^{4} + 66596 T^{5} + 337064 T^{6} + 4553292 T^{7} - 29366017 T^{8} - 645465440 T^{9} - 3441343568 T^{10} - 51311871904 T^{11} - 755349580500 T^{12} - 51311871904 p T^{13} - 3441343568 p^{2} T^{14} - 645465440 p^{3} T^{15} - 29366017 p^{4} T^{16} + 4553292 p^{5} T^{17} + 337064 p^{6} T^{18} + 66596 p^{7} T^{19} + 15698 p^{8} T^{20} + 1156 p^{9} T^{21} + 72 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 + 32 T + 512 T^{2} + 5744 T^{3} + 39850 T^{4} + 51440 T^{5} - 2260352 T^{6} - 34177952 T^{7} - 265415825 T^{8} - 630997504 T^{9} + 10780889984 T^{10} + 203619700384 T^{11} + 2167679393548 T^{12} + 203619700384 p T^{13} + 10780889984 p^{2} T^{14} - 630997504 p^{3} T^{15} - 265415825 p^{4} T^{16} - 34177952 p^{5} T^{17} - 2260352 p^{6} T^{18} + 51440 p^{7} T^{19} + 39850 p^{8} T^{20} + 5744 p^{9} T^{21} + 512 p^{10} T^{22} + 32 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( ( 1 - 12 T + 366 T^{2} - 3292 T^{3} + 60143 T^{4} - 434224 T^{5} + 5969684 T^{6} - 434224 p T^{7} + 60143 p^{2} T^{8} - 3292 p^{3} T^{9} + 366 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 83 | \( 1 - 1152 T^{3} - 14018 T^{4} + 107208 T^{5} + 663552 T^{6} + 16643448 T^{7} - 66730753 T^{8} - 1415095272 T^{9} - 598752 p^{2} T^{10} + 17426520 p T^{11} + 1627282984756 T^{12} + 17426520 p^{2} T^{13} - 598752 p^{4} T^{14} - 1415095272 p^{3} T^{15} - 66730753 p^{4} T^{16} + 16643448 p^{5} T^{17} + 663552 p^{6} T^{18} + 107208 p^{7} T^{19} - 14018 p^{8} T^{20} - 1152 p^{9} T^{21} + p^{12} T^{24} \) | |
| 89 | \( 1 - 4 T + 8 T^{2} + 220 T^{3} + 4694 T^{4} - 27692 T^{5} + 97416 T^{6} - 689244 T^{7} + 36691711 T^{8} + 158870528 T^{9} - 1154496752 T^{10} - 8032693040 T^{11} + 990018630692 T^{12} - 8032693040 p T^{13} - 1154496752 p^{2} T^{14} + 158870528 p^{3} T^{15} + 36691711 p^{4} T^{16} - 689244 p^{5} T^{17} + 97416 p^{6} T^{18} - 27692 p^{7} T^{19} + 4694 p^{8} T^{20} + 220 p^{9} T^{21} + 8 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 - 208 T^{3} + 19658 T^{4} - 5264 T^{5} + 21632 T^{6} - 6420224 T^{7} + 95827919 T^{8} - 24422528 T^{9} + 924019584 T^{10} - 39390521632 T^{11} + 21692174028 T^{12} - 39390521632 p T^{13} + 924019584 p^{2} T^{14} - 24422528 p^{3} T^{15} + 95827919 p^{4} T^{16} - 6420224 p^{5} T^{17} + 21632 p^{6} T^{18} - 5264 p^{7} T^{19} + 19658 p^{8} T^{20} - 208 p^{9} T^{21} + p^{12} T^{24} \) | |
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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
−4.16582696060404465572652368070, −3.97801204225206324104780719780, −3.92761739634019657088674697262, −3.89274651185488029119983881256, −3.45085175815979316852245023718, −3.32301798039864515144177855881, −3.26428956322092527129987732744, −3.13872881873728489305825042624, −2.92139288424877414704109205086, −2.87240705825164789078016219131, −2.86553277570865329120091879067, −2.86072800290426622591154756869, −2.77461530427877415836363093137, −2.49975275448303117950539589522, −2.32752590306520533766534334033, −2.27143444553231705896842230549, −2.18480086229547264965400110369, −1.99939164197651085363288669054, −1.80775590129095974946344862483, −1.55961680533240204658414938615, −1.53906617068432650645045189570, −1.44389473807691399424555585497, −1.00401231021248894868391693601, −0.915110704236847474709858923405, −0.15816066960336783384088723287, 0.15816066960336783384088723287, 0.915110704236847474709858923405, 1.00401231021248894868391693601, 1.44389473807691399424555585497, 1.53906617068432650645045189570, 1.55961680533240204658414938615, 1.80775590129095974946344862483, 1.99939164197651085363288669054, 2.18480086229547264965400110369, 2.27143444553231705896842230549, 2.32752590306520533766534334033, 2.49975275448303117950539589522, 2.77461530427877415836363093137, 2.86072800290426622591154756869, 2.86553277570865329120091879067, 2.87240705825164789078016219131, 2.92139288424877414704109205086, 3.13872881873728489305825042624, 3.26428956322092527129987732744, 3.32301798039864515144177855881, 3.45085175815979316852245023718, 3.89274651185488029119983881256, 3.92761739634019657088674697262, 3.97801204225206324104780719780, 4.16582696060404465572652368070
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.