Dirichlet series
| L(s) = 1 | − 2·3-s + 3·4-s + 6·5-s + 8·7-s + 5·9-s − 12·11-s − 6·12-s − 12·15-s + 3·16-s + 12·17-s + 18·20-s − 16·21-s − 24·23-s + 15·25-s − 10·27-s + 24·28-s + 12·31-s + 24·33-s + 48·35-s + 15·36-s − 8·37-s − 4·41-s − 36·44-s + 30·45-s + 16·47-s − 6·48-s + 25·49-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 3/2·4-s + 2.68·5-s + 3.02·7-s + 5/3·9-s − 3.61·11-s − 1.73·12-s − 3.09·15-s + 3/4·16-s + 2.91·17-s + 4.02·20-s − 3.49·21-s − 5.00·23-s + 3·25-s − 1.92·27-s + 4.53·28-s + 2.15·31-s + 4.17·33-s + 8.11·35-s + 5/2·36-s − 1.31·37-s − 0.624·41-s − 5.42·44-s + 4.47·45-s + 2.33·47-s − 0.866·48-s + 25/7·49-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 5^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 5^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 3^{12} \cdot 5^{12} \cdot 7^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(494.261\) |
| Root analytic conductor: | \(1.29493\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{12} \cdot 3^{12} \cdot 5^{12} \cdot 7^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(5.820696155\) |
| \(L(\frac12)\) | \(\approx\) | \(5.820696155\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T^{2} + T^{4} )^{3} \) |
| 3 | \( 1 + 2 T - T^{2} - 2 T^{3} - 8 T^{4} - 2 p T^{5} + 11 p T^{6} - 2 p^{2} T^{7} - 8 p^{2} T^{8} - 2 p^{3} T^{9} - p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \) | |
| 5 | \( ( 1 - T + T^{2} )^{6} \) | |
| 7 | \( 1 - 8 T + 39 T^{2} - 152 T^{3} + 69 p T^{4} - 1360 T^{5} + 3642 T^{6} - 1360 p T^{7} + 69 p^{3} T^{8} - 152 p^{3} T^{9} + 39 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \) | |
| good | 11 | \( 1 + 12 T + 104 T^{2} + 672 T^{3} + 3656 T^{4} + 16524 T^{5} + 63276 T^{6} + 198252 T^{7} + 468664 T^{8} + 516768 T^{9} - 2129576 T^{10} - 16995156 T^{11} - 68897930 T^{12} - 16995156 p T^{13} - 2129576 p^{2} T^{14} + 516768 p^{3} T^{15} + 468664 p^{4} T^{16} + 198252 p^{5} T^{17} + 63276 p^{6} T^{18} + 16524 p^{7} T^{19} + 3656 p^{8} T^{20} + 672 p^{9} T^{21} + 104 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \) |
| 13 | \( 1 - 24 T^{2} + 632 T^{4} - 13172 T^{6} + 225768 T^{8} - 3379208 T^{10} + 49928150 T^{12} - 3379208 p^{2} T^{14} + 225768 p^{4} T^{16} - 13172 p^{6} T^{18} + 632 p^{8} T^{20} - 24 p^{10} T^{22} + p^{12} T^{24} \) | |
| 17 | \( 1 - 12 T + 10 T^{2} + 240 T^{3} + 929 T^{4} - 10248 T^{5} - 21642 T^{6} + 175188 T^{7} + 650254 T^{8} - 1893108 T^{9} - 17461318 T^{10} + 1450776 p T^{11} + 233538997 T^{12} + 1450776 p^{2} T^{13} - 17461318 p^{2} T^{14} - 1893108 p^{3} T^{15} + 650254 p^{4} T^{16} + 175188 p^{5} T^{17} - 21642 p^{6} T^{18} - 10248 p^{7} T^{19} + 929 p^{8} T^{20} + 240 p^{9} T^{21} + 10 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 + 24 T^{2} - 316 T^{4} + 144 T^{5} - 5636 T^{6} - 33888 T^{7} + 181380 T^{8} - 941760 T^{9} + 1187824 T^{10} + 9129072 T^{11} - 54892570 T^{12} + 9129072 p T^{13} + 1187824 p^{2} T^{14} - 941760 p^{3} T^{15} + 181380 p^{4} T^{16} - 33888 p^{5} T^{17} - 5636 p^{6} T^{18} + 144 p^{7} T^{19} - 316 p^{8} T^{20} + 24 p^{10} T^{22} + p^{12} T^{24} \) | |
| 23 | \( 1 + 24 T + 367 T^{2} + 4200 T^{3} + 39535 T^{4} + 13992 p T^{5} + 2347952 T^{6} + 15699792 T^{7} + 97859137 T^{8} + 573159912 T^{9} + 3169289665 T^{10} + 16552915872 T^{11} + 81697151198 T^{12} + 16552915872 p T^{13} + 3169289665 p^{2} T^{14} + 573159912 p^{3} T^{15} + 97859137 p^{4} T^{16} + 15699792 p^{5} T^{17} + 2347952 p^{6} T^{18} + 13992 p^{8} T^{19} + 39535 p^{8} T^{20} + 4200 p^{9} T^{21} + 367 p^{10} T^{22} + 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 - 158 T^{2} + 12415 T^{4} - 625726 T^{6} + 22774147 T^{8} - 673287380 T^{10} + 19008498890 T^{12} - 673287380 p^{2} T^{14} + 22774147 p^{4} T^{16} - 625726 p^{6} T^{18} + 12415 p^{8} T^{20} - 158 p^{10} T^{22} + p^{12} T^{24} \) | |
| 31 | \( 1 - 12 T + 162 T^{2} - 1368 T^{3} + 10769 T^{4} - 77664 T^{5} + 554878 T^{6} - 3968772 T^{7} + 891330 p T^{8} - 178188132 T^{9} + 1059815698 T^{10} - 6109630032 T^{11} + 33269854325 T^{12} - 6109630032 p T^{13} + 1059815698 p^{2} T^{14} - 178188132 p^{3} T^{15} + 891330 p^{5} T^{16} - 3968772 p^{5} T^{17} + 554878 p^{6} T^{18} - 77664 p^{7} T^{19} + 10769 p^{8} T^{20} - 1368 p^{9} T^{21} + 162 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 + 8 T - 104 T^{2} - 832 T^{3} + 6788 T^{4} + 46312 T^{5} - 8812 p T^{6} - 1583912 T^{7} + 13285604 T^{8} + 33469952 T^{9} - 514631840 T^{10} - 373683016 T^{11} + 19047573830 T^{12} - 373683016 p T^{13} - 514631840 p^{2} T^{14} + 33469952 p^{3} T^{15} + 13285604 p^{4} T^{16} - 1583912 p^{5} T^{17} - 8812 p^{7} T^{18} + 46312 p^{7} T^{19} + 6788 p^{8} T^{20} - 832 p^{9} T^{21} - 104 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( ( 1 + 2 T + 101 T^{2} + 150 T^{3} + 6062 T^{4} + 7442 T^{5} + 295913 T^{6} + 7442 p T^{7} + 6062 p^{2} T^{8} + 150 p^{3} T^{9} + 101 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 43 | \( ( 1 + 131 T^{2} - 196 T^{3} + 7479 T^{4} - 24892 T^{5} + 314798 T^{6} - 24892 p T^{7} + 7479 p^{2} T^{8} - 196 p^{3} T^{9} + 131 p^{4} T^{10} + p^{6} T^{12} )^{2} \) | |
| 47 | \( 1 - 16 T + 32 T^{2} + 544 T^{3} + 428 T^{4} - 39000 T^{5} + 312596 T^{6} - 2377608 T^{7} + 2268284 T^{8} + 67449856 T^{9} + 407948184 T^{10} - 5931851872 T^{11} + 27380961750 T^{12} - 5931851872 p T^{13} + 407948184 p^{2} T^{14} + 67449856 p^{3} T^{15} + 2268284 p^{4} T^{16} - 2377608 p^{5} T^{17} + 312596 p^{6} T^{18} - 39000 p^{7} T^{19} + 428 p^{8} T^{20} + 544 p^{9} T^{21} + 32 p^{10} T^{22} - 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 + 48 T + 24 p T^{2} + 24192 T^{3} + 372104 T^{4} + 4941024 T^{5} + 58639988 T^{6} + 632702016 T^{7} + 6267491832 T^{8} + 57492176256 T^{9} + 491936676296 T^{10} + 3943130126256 T^{11} + 29642342340470 T^{12} + 3943130126256 p T^{13} + 491936676296 p^{2} T^{14} + 57492176256 p^{3} T^{15} + 6267491832 p^{4} T^{16} + 632702016 p^{5} T^{17} + 58639988 p^{6} T^{18} + 4941024 p^{7} T^{19} + 372104 p^{8} T^{20} + 24192 p^{9} T^{21} + 24 p^{11} T^{22} + 48 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 - 12 T - 62 T^{2} + 192 T^{3} + 13661 T^{4} - 4488 T^{5} - 291210 T^{6} - 10035204 T^{7} + 17708314 T^{8} + 439140108 T^{9} + 4538367434 T^{10} - 27175198584 T^{11} - 197381258843 T^{12} - 27175198584 p T^{13} + 4538367434 p^{2} T^{14} + 439140108 p^{3} T^{15} + 17708314 p^{4} T^{16} - 10035204 p^{5} T^{17} - 291210 p^{6} T^{18} - 4488 p^{7} T^{19} + 13661 p^{8} T^{20} + 192 p^{9} T^{21} - 62 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 + 30 T + 633 T^{2} + 9990 T^{3} + 134066 T^{4} + 1612650 T^{5} + 18045535 T^{6} + 51210 p^{2} T^{7} + 1899056448 T^{8} + 17814782910 T^{9} + 157346567941 T^{10} + 1317498445782 T^{11} + 10515537066752 T^{12} + 1317498445782 p T^{13} + 157346567941 p^{2} T^{14} + 17814782910 p^{3} T^{15} + 1899056448 p^{4} T^{16} + 51210 p^{7} T^{17} + 18045535 p^{6} T^{18} + 1612650 p^{7} T^{19} + 134066 p^{8} T^{20} + 9990 p^{9} T^{21} + 633 p^{10} T^{22} + 30 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 + 4 T - 339 T^{2} - 1212 T^{3} + 65398 T^{4} + 197860 T^{5} - 8949237 T^{6} - 19422588 T^{7} + 965230072 T^{8} + 1219402116 T^{9} - 85058357911 T^{10} - 32989889052 T^{11} + 6246904001980 T^{12} - 32989889052 p T^{13} - 85058357911 p^{2} T^{14} + 1219402116 p^{3} T^{15} + 965230072 p^{4} T^{16} - 19422588 p^{5} T^{17} - 8949237 p^{6} T^{18} + 197860 p^{7} T^{19} + 65398 p^{8} T^{20} - 1212 p^{9} T^{21} - 339 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 - 300 T^{2} + 49610 T^{4} - 6300412 T^{6} + 667955583 T^{8} - 59469293656 T^{10} + 4533955360172 T^{12} - 59469293656 p^{2} T^{14} + 667955583 p^{4} T^{16} - 6300412 p^{6} T^{18} + 49610 p^{8} T^{20} - 300 p^{10} T^{22} + p^{12} T^{24} \) | |
| 73 | \( 1 + 158 T^{2} + 8161 T^{4} - 137736 T^{5} + 413890 T^{6} - 21056400 T^{7} + 34702222 T^{8} - 1012533192 T^{9} + 9774206798 T^{10} - 39652148664 T^{11} + 1268896234901 T^{12} - 39652148664 p T^{13} + 9774206798 p^{2} T^{14} - 1012533192 p^{3} T^{15} + 34702222 p^{4} T^{16} - 21056400 p^{5} T^{17} + 413890 p^{6} T^{18} - 137736 p^{7} T^{19} + 8161 p^{8} T^{20} + 158 p^{10} T^{22} + p^{12} T^{24} \) | |
| 79 | \( 1 + 4 T - 254 T^{2} + 1984 T^{3} + 51521 T^{4} - 543784 T^{5} - 2546434 T^{6} + 103559324 T^{7} - 209408530 T^{8} - 8333233172 T^{9} + 86160957682 T^{10} + 365142325960 T^{11} - 8321189358475 T^{12} + 365142325960 p T^{13} + 86160957682 p^{2} T^{14} - 8333233172 p^{3} T^{15} - 209408530 p^{4} T^{16} + 103559324 p^{5} T^{17} - 2546434 p^{6} T^{18} - 543784 p^{7} T^{19} + 51521 p^{8} T^{20} + 1984 p^{9} T^{21} - 254 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( ( 1 + 20 T + 635 T^{2} + 8628 T^{3} + 148691 T^{4} + 1458128 T^{5} + 17076098 T^{6} + 1458128 p T^{7} + 148691 p^{2} T^{8} + 8628 p^{3} T^{9} + 635 p^{4} T^{10} + 20 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 89 | \( 1 - 26 T + 101 T^{2} + 1838 T^{3} + 1058 T^{4} - 234822 T^{5} - 191269 T^{6} + 17352042 T^{7} - 36283792 T^{8} - 94067578 T^{9} - 6641487 T^{10} - 7507089026 T^{11} - 13384168368 T^{12} - 7507089026 p T^{13} - 6641487 p^{2} T^{14} - 94067578 p^{3} T^{15} - 36283792 p^{4} T^{16} + 17352042 p^{5} T^{17} - 191269 p^{6} T^{18} - 234822 p^{7} T^{19} + 1058 p^{8} T^{20} + 1838 p^{9} T^{21} + 101 p^{10} T^{22} - 26 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 - 484 T^{2} + 116170 T^{4} - 18891668 T^{6} + 2452131295 T^{8} - 279695423560 T^{10} + 28680693351980 T^{12} - 279695423560 p^{2} T^{14} + 2452131295 p^{4} T^{16} - 18891668 p^{6} T^{18} + 116170 p^{8} T^{20} - 484 p^{10} T^{22} + p^{12} T^{24} \) | |
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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
−4.33573602666669652503405282060, −4.26770060976838893375887947455, −4.24833499223080492881474288933, −4.16986536875118829197416577030, −3.81366168170160508571584095722, −3.78418298920647880296727136416, −3.57227045484527103478670516501, −3.55060029447280724331197076179, −3.17666266474230193120725678557, −3.15557461016795787422721657125, −2.98969038829316184014738805665, −2.98391804674177818981037346196, −2.70824443451765279128576889599, −2.65730405754213712277913891424, −2.49821820845109492512498500137, −2.42933429068951908551094894458, −1.91449371543021549603218203946, −1.86477779994935201303050253920, −1.77737465197331807025572205023, −1.77306814410922867735843027912, −1.69975663870794962268082063571, −1.69465962564745372882092565780, −1.58762699854173022032918645967, −0.791014032461784197469503130700, −0.61171406472908033165882173892, 0.61171406472908033165882173892, 0.791014032461784197469503130700, 1.58762699854173022032918645967, 1.69465962564745372882092565780, 1.69975663870794962268082063571, 1.77306814410922867735843027912, 1.77737465197331807025572205023, 1.86477779994935201303050253920, 1.91449371543021549603218203946, 2.42933429068951908551094894458, 2.49821820845109492512498500137, 2.65730405754213712277913891424, 2.70824443451765279128576889599, 2.98391804674177818981037346196, 2.98969038829316184014738805665, 3.15557461016795787422721657125, 3.17666266474230193120725678557, 3.55060029447280724331197076179, 3.57227045484527103478670516501, 3.78418298920647880296727136416, 3.81366168170160508571584095722, 4.16986536875118829197416577030, 4.24833499223080492881474288933, 4.26770060976838893375887947455, 4.33573602666669652503405282060
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.