Dirichlet series
| L(s) = 1 | − 6·2-s + 15·4-s − 2·5-s − 2·7-s − 14·8-s + 3·9-s + 12·10-s + 6·11-s − 8·13-s + 12·14-s − 21·16-s + 18·17-s − 18·18-s − 6·19-s − 30·20-s − 36·22-s + 6·23-s − 3·25-s + 48·26-s − 30·28-s + 14·29-s + 84·32-s − 108·34-s + 4·35-s + 45·36-s − 12·37-s + 36·38-s + ⋯ |
| L(s) = 1 | − 4.24·2-s + 15/2·4-s − 0.894·5-s − 0.755·7-s − 4.94·8-s + 9-s + 3.79·10-s + 1.80·11-s − 2.21·13-s + 3.20·14-s − 5.25·16-s + 4.36·17-s − 4.24·18-s − 1.37·19-s − 6.70·20-s − 7.67·22-s + 1.25·23-s − 3/5·25-s + 9.41·26-s − 5.66·28-s + 2.59·29-s + 14.8·32-s − 18.5·34-s + 0.676·35-s + 15/2·36-s − 1.97·37-s + 5.83·38-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 5^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 5^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 3^{12} \cdot 5^{12} \cdot 13^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(831956.\) |
| Root analytic conductor: | \(1.76469\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{12} \cdot 3^{12} \cdot 5^{12} \cdot 13^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.02084236325\) |
| \(L(\frac12)\) | \(\approx\) | \(0.02084236325\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T + T^{2} )^{6} \) |
| 3 | \( ( 1 - T^{2} + T^{4} )^{3} \) | |
| 5 | \( 1 + 2 T + 7 T^{2} + 6 T^{3} + 3 p T^{4} + 32 T^{5} + 114 T^{6} + 32 p T^{7} + 3 p^{3} T^{8} + 6 p^{3} T^{9} + 7 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \) | |
| 13 | \( 1 + 8 T + 36 T^{2} + 40 T^{3} - 232 T^{4} - 2040 T^{5} - 7838 T^{6} - 2040 p T^{7} - 232 p^{2} T^{8} + 40 p^{3} T^{9} + 36 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \) | |
| good | 7 | \( 1 + 2 T - 23 T^{2} - 62 T^{3} + 254 T^{4} + 874 T^{5} - 1749 T^{6} - 1154 p T^{7} + 7052 T^{8} + 51162 T^{9} + 7565 T^{10} - 150326 T^{11} - 259320 T^{12} - 150326 p T^{13} + 7565 p^{2} T^{14} + 51162 p^{3} T^{15} + 7052 p^{4} T^{16} - 1154 p^{6} T^{17} - 1749 p^{6} T^{18} + 874 p^{7} T^{19} + 254 p^{8} T^{20} - 62 p^{9} T^{21} - 23 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \) |
| 11 | \( 1 - 6 T + 64 T^{2} - 312 T^{3} + 2056 T^{4} - 9354 T^{5} + 4304 p T^{6} - 200514 T^{7} + 76792 p T^{8} - 3316248 T^{9} + 12268912 T^{10} - 44392686 T^{11} + 147915278 T^{12} - 44392686 p T^{13} + 12268912 p^{2} T^{14} - 3316248 p^{3} T^{15} + 76792 p^{5} T^{16} - 200514 p^{5} T^{17} + 4304 p^{7} T^{18} - 9354 p^{7} T^{19} + 2056 p^{8} T^{20} - 312 p^{9} T^{21} + 64 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 - 18 T + 197 T^{2} - 1602 T^{3} + 10786 T^{4} - 63078 T^{5} + 334331 T^{6} - 1636158 T^{7} + 443776 p T^{8} - 33083826 T^{9} + 139928849 T^{10} - 578245362 T^{11} + 2383071680 T^{12} - 578245362 p T^{13} + 139928849 p^{2} T^{14} - 33083826 p^{3} T^{15} + 443776 p^{5} T^{16} - 1636158 p^{5} T^{17} + 334331 p^{6} T^{18} - 63078 p^{7} T^{19} + 10786 p^{8} T^{20} - 1602 p^{9} T^{21} + 197 p^{10} T^{22} - 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 + 6 T + 61 T^{2} + 294 T^{3} + 1542 T^{4} + 5790 T^{5} + 19223 T^{6} + 78270 T^{7} + 263444 T^{8} + 1783974 T^{9} + 8840433 T^{10} + 51684582 T^{11} + 215919880 T^{12} + 51684582 p T^{13} + 8840433 p^{2} T^{14} + 1783974 p^{3} T^{15} + 263444 p^{4} T^{16} + 78270 p^{5} T^{17} + 19223 p^{6} T^{18} + 5790 p^{7} T^{19} + 1542 p^{8} T^{20} + 294 p^{9} T^{21} + 61 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 - 6 T + 45 T^{2} - 198 T^{3} + 610 T^{4} + 4074 T^{5} - 1715 p T^{6} + 229650 T^{7} - 1158136 T^{8} + 2505762 T^{9} + 9180361 T^{10} - 132212382 T^{11} + 612206888 T^{12} - 132212382 p T^{13} + 9180361 p^{2} T^{14} + 2505762 p^{3} T^{15} - 1158136 p^{4} T^{16} + 229650 p^{5} T^{17} - 1715 p^{7} T^{18} + 4074 p^{7} T^{19} + 610 p^{8} T^{20} - 198 p^{9} T^{21} + 45 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 - 14 T + 16 T^{2} + 664 T^{3} - 2664 T^{4} - 11782 T^{5} + 47832 T^{6} + 296622 T^{7} + 89240 T^{8} - 14376232 T^{9} + 24101024 T^{10} + 238729942 T^{11} - 1595438210 T^{12} + 238729942 p T^{13} + 24101024 p^{2} T^{14} - 14376232 p^{3} T^{15} + 89240 p^{4} T^{16} + 296622 p^{5} T^{17} + 47832 p^{6} T^{18} - 11782 p^{7} T^{19} - 2664 p^{8} T^{20} + 664 p^{9} T^{21} + 16 p^{10} T^{22} - 14 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 - 174 T^{2} + 17479 T^{4} - 1226254 T^{6} + 65688635 T^{8} - 2791262084 T^{10} + 95948057978 T^{12} - 2791262084 p^{2} T^{14} + 65688635 p^{4} T^{16} - 1226254 p^{6} T^{18} + 17479 p^{8} T^{20} - 174 p^{10} T^{22} + p^{12} T^{24} \) | |
| 37 | \( 1 + 12 T - 13 T^{2} - 748 T^{3} - 2473 T^{4} + 10272 T^{5} + 95952 T^{6} + 273816 T^{7} - 1904591 T^{8} - 33447492 T^{9} - 139069523 T^{10} + 954632332 T^{11} + 12858316046 T^{12} + 954632332 p T^{13} - 139069523 p^{2} T^{14} - 33447492 p^{3} T^{15} - 1904591 p^{4} T^{16} + 273816 p^{5} T^{17} + 95952 p^{6} T^{18} + 10272 p^{7} T^{19} - 2473 p^{8} T^{20} - 748 p^{9} T^{21} - 13 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 + 18 T + 341 T^{2} + 4194 T^{3} + 50482 T^{4} + 499830 T^{5} + 4801691 T^{6} + 40879182 T^{7} + 337655168 T^{8} + 2548727826 T^{9} + 18713647169 T^{10} + 127391729010 T^{11} + 845219074352 T^{12} + 127391729010 p T^{13} + 18713647169 p^{2} T^{14} + 2548727826 p^{3} T^{15} + 337655168 p^{4} T^{16} + 40879182 p^{5} T^{17} + 4801691 p^{6} T^{18} + 499830 p^{7} T^{19} + 50482 p^{8} T^{20} + 4194 p^{9} T^{21} + 341 p^{10} T^{22} + 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 + 36 T + 793 T^{2} + 12996 T^{3} + 173394 T^{4} + 1956084 T^{5} + 447185 p T^{6} + 167806644 T^{7} + 1324093004 T^{8} + 9621159300 T^{9} + 65921206941 T^{10} + 436765148388 T^{11} + 2865309355372 T^{12} + 436765148388 p T^{13} + 65921206941 p^{2} T^{14} + 9621159300 p^{3} T^{15} + 1324093004 p^{4} T^{16} + 167806644 p^{5} T^{17} + 447185 p^{7} T^{18} + 1956084 p^{7} T^{19} + 173394 p^{8} T^{20} + 12996 p^{9} T^{21} + 793 p^{10} T^{22} + 36 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( ( 1 - 8 T + 184 T^{2} - 600 T^{3} + 10428 T^{4} + 6112 T^{5} + 380922 T^{6} + 6112 p T^{7} + 10428 p^{2} T^{8} - 600 p^{3} T^{9} + 184 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 53 | \( 1 - 128 T^{2} + 3592 T^{4} + 165508 T^{6} - 2245288 T^{8} - 1080283184 T^{10} + 86360756726 T^{12} - 1080283184 p^{2} T^{14} - 2245288 p^{4} T^{16} + 165508 p^{6} T^{18} + 3592 p^{8} T^{20} - 128 p^{10} T^{22} + p^{12} T^{24} \) | |
| 59 | \( 1 + 36 T + 813 T^{2} + 13716 T^{3} + 186858 T^{4} + 2135412 T^{5} + 21101127 T^{6} + 182816100 T^{7} + 1417839948 T^{8} + 10053293508 T^{9} + 67598064417 T^{10} + 455934803796 T^{11} + 3321686461700 T^{12} + 455934803796 p T^{13} + 67598064417 p^{2} T^{14} + 10053293508 p^{3} T^{15} + 1417839948 p^{4} T^{16} + 182816100 p^{5} T^{17} + 21101127 p^{6} T^{18} + 2135412 p^{7} T^{19} + 186858 p^{8} T^{20} + 13716 p^{9} T^{21} + 813 p^{10} T^{22} + 36 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 - 10 T - 231 T^{2} + 1678 T^{3} + 38122 T^{4} - 170302 T^{5} - 4572793 T^{6} + 12436962 T^{7} + 423313480 T^{8} - 601350602 T^{9} - 32798192891 T^{10} + 13818430974 T^{11} + 2165548491568 T^{12} + 13818430974 p T^{13} - 32798192891 p^{2} T^{14} - 601350602 p^{3} T^{15} + 423313480 p^{4} T^{16} + 12436962 p^{5} T^{17} - 4572793 p^{6} T^{18} - 170302 p^{7} T^{19} + 38122 p^{8} T^{20} + 1678 p^{9} T^{21} - 231 p^{10} T^{22} - 10 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 - 4 T - 238 T^{2} + 1856 T^{3} + 28365 T^{4} - 320664 T^{5} - 1684986 T^{6} + 34626164 T^{7} + 2659226 T^{8} - 2274090556 T^{9} + 10436172442 T^{10} + 66052009432 T^{11} - 1013395621067 T^{12} + 66052009432 p T^{13} + 10436172442 p^{2} T^{14} - 2274090556 p^{3} T^{15} + 2659226 p^{4} T^{16} + 34626164 p^{5} T^{17} - 1684986 p^{6} T^{18} - 320664 p^{7} T^{19} + 28365 p^{8} T^{20} + 1856 p^{9} T^{21} - 238 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 + 12 T + 254 T^{2} + 2472 T^{3} + 34021 T^{4} + 276816 T^{5} + 2214938 T^{6} + 11329380 T^{7} - 4931254 T^{8} - 805549452 T^{9} - 16812280330 T^{10} - 166476163968 T^{11} - 1736803148659 T^{12} - 166476163968 p T^{13} - 16812280330 p^{2} T^{14} - 805549452 p^{3} T^{15} - 4931254 p^{4} T^{16} + 11329380 p^{5} T^{17} + 2214938 p^{6} T^{18} + 276816 p^{7} T^{19} + 34021 p^{8} T^{20} + 2472 p^{9} T^{21} + 254 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( ( 1 - 14 T + 175 T^{2} - 2042 T^{3} + 21419 T^{4} - 188408 T^{5} + 1809530 T^{6} - 188408 p T^{7} + 21419 p^{2} T^{8} - 2042 p^{3} T^{9} + 175 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 79 | \( ( 1 - 2 T + 295 T^{2} - 2 T^{3} + 40943 T^{4} + 49360 T^{5} + 3743570 T^{6} + 49360 p T^{7} + 40943 p^{2} T^{8} - 2 p^{3} T^{9} + 295 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 83 | \( ( 1 - 36 T + 858 T^{2} - 14796 T^{3} + 206295 T^{4} - 2392344 T^{5} + 23560108 T^{6} - 2392344 p T^{7} + 206295 p^{2} T^{8} - 14796 p^{3} T^{9} + 858 p^{4} T^{10} - 36 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 89 | \( 1 - 18 T + 225 T^{2} - 2106 T^{3} + 13930 T^{4} - 69414 T^{5} + 500783 T^{6} - 13413294 T^{7} + 192067808 T^{8} - 23896218 p T^{9} + 16430373661 T^{10} - 924473106 p T^{11} + 513578497736 T^{12} - 924473106 p^{2} T^{13} + 16430373661 p^{2} T^{14} - 23896218 p^{4} T^{15} + 192067808 p^{4} T^{16} - 13413294 p^{5} T^{17} + 500783 p^{6} T^{18} - 69414 p^{7} T^{19} + 13930 p^{8} T^{20} - 2106 p^{9} T^{21} + 225 p^{10} T^{22} - 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 + 48 T + 1058 T^{2} + 14080 T^{3} + 118085 T^{4} + 316224 T^{5} - 8738586 T^{6} - 165823248 T^{7} - 1532156150 T^{8} - 7802628048 T^{9} + 6109396906 T^{10} + 716778222080 T^{11} + 9694213700813 T^{12} + 716778222080 p T^{13} + 6109396906 p^{2} T^{14} - 7802628048 p^{3} T^{15} - 1532156150 p^{4} T^{16} - 165823248 p^{5} T^{17} - 8738586 p^{6} T^{18} + 316224 p^{7} T^{19} + 118085 p^{8} T^{20} + 14080 p^{9} T^{21} + 1058 p^{10} T^{22} + 48 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−3.74696942820947032660676397892, −3.73984252976089014691064132818, −3.61222649285003970088733692278, −3.39480600873837809870063669586, −3.29790949815451012322332123848, −3.22307805686509252698031940310, −3.15836002058783795694549406915, −3.15054218299451989751892011311, −3.09196715857896594389506906372, −2.66602949602715065844138439340, −2.45638863734611156278439058664, −2.39315599782289821588181289753, −2.36830862471692709465186895584, −2.08765456664668791881546106255, −1.94149876417019315933118675496, −1.90799401330257764817075776668, −1.84843441168044560255644499147, −1.41903600236174698500313831235, −1.32726594854625284861379528582, −1.29918457647394720211369090460, −1.00743015795294230098171818119, −0.990231505264190338223272438423, −0.837154255184171988967965811862, −0.44522657724787774130968135510, −0.090665345169537599148485362180, 0.090665345169537599148485362180, 0.44522657724787774130968135510, 0.837154255184171988967965811862, 0.990231505264190338223272438423, 1.00743015795294230098171818119, 1.29918457647394720211369090460, 1.32726594854625284861379528582, 1.41903600236174698500313831235, 1.84843441168044560255644499147, 1.90799401330257764817075776668, 1.94149876417019315933118675496, 2.08765456664668791881546106255, 2.36830862471692709465186895584, 2.39315599782289821588181289753, 2.45638863734611156278439058664, 2.66602949602715065844138439340, 3.09196715857896594389506906372, 3.15054218299451989751892011311, 3.15836002058783795694549406915, 3.22307805686509252698031940310, 3.29790949815451012322332123848, 3.39480600873837809870063669586, 3.61222649285003970088733692278, 3.73984252976089014691064132818, 3.74696942820947032660676397892
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.