Dirichlet series
| L(s) = 1 | − 4·2-s + 6·4-s − 4·8-s + 16-s + 24·25-s − 32·29-s + 16·31-s + 32·37-s − 96·50-s + 32·53-s + 128·58-s + 16·59-s − 64·62-s − 128·74-s + 16·83-s + 144·100-s − 16·103-s − 128·106-s + 16·109-s + 64·113-s − 192·116-s − 64·118-s + 28·121-s + 96·124-s + 127-s + 8·128-s + 131-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 3·4-s − 1.41·8-s + 1/4·16-s + 24/5·25-s − 5.94·29-s + 2.87·31-s + 5.26·37-s − 13.5·50-s + 4.39·53-s + 16.8·58-s + 2.08·59-s − 8.12·62-s − 14.8·74-s + 1.75·83-s + 72/5·100-s − 1.57·103-s − 12.4·106-s + 1.53·109-s + 6.02·113-s − 17.8·116-s − 5.89·118-s + 2.54·121-s + 8.62·124-s + 0.0887·127-s + 0.707·128-s + 0.0873·131-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{24} \cdot 3^{24} \cdot 7^{24}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.09969\times 10^{13}\) |
| Root analytic conductor: | \(3.75308\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{24} \cdot 3^{24} \cdot 7^{24} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.8822829851\) |
| \(L(\frac12)\) | \(\approx\) | \(0.8822829851\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 + p^{2} T + 5 p T^{2} + 5 p^{2} T^{3} + 35 T^{4} + 7 p^{3} T^{5} + 21 p^{2} T^{6} + 7 p^{4} T^{7} + 35 p^{2} T^{8} + 5 p^{5} T^{9} + 5 p^{5} T^{10} + p^{7} T^{11} + p^{6} T^{12} \) |
| 3 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 5 | \( 1 - 24 T^{2} + 296 T^{4} - 2744 T^{6} + 21123 T^{8} - 134032 T^{10} + 718928 T^{12} - 134032 p^{2} T^{14} + 21123 p^{4} T^{16} - 2744 p^{6} T^{18} + 296 p^{8} T^{20} - 24 p^{10} T^{22} + p^{12} T^{24} \) |
| 11 | \( 1 - 28 T^{2} + 586 T^{4} - 9388 T^{6} + 131503 T^{8} - 1628696 T^{10} + 19283052 T^{12} - 1628696 p^{2} T^{14} + 131503 p^{4} T^{16} - 9388 p^{6} T^{18} + 586 p^{8} T^{20} - 28 p^{10} T^{22} + p^{12} T^{24} \) | |
| 13 | \( 1 - 64 T^{2} + 176 p T^{4} - 59968 T^{6} + 1236915 T^{8} - 20973696 T^{10} + 297955296 T^{12} - 20973696 p^{2} T^{14} + 1236915 p^{4} T^{16} - 59968 p^{6} T^{18} + 176 p^{9} T^{20} - 64 p^{10} T^{22} + p^{12} T^{24} \) | |
| 17 | \( 1 - 88 T^{2} + 4536 T^{4} - 165688 T^{6} + 4700147 T^{8} - 107164048 T^{10} + 2007265136 T^{12} - 107164048 p^{2} T^{14} + 4700147 p^{4} T^{16} - 165688 p^{6} T^{18} + 4536 p^{8} T^{20} - 88 p^{10} T^{22} + p^{12} T^{24} \) | |
| 19 | \( ( 1 + 42 T^{2} - 32 T^{3} + 1231 T^{4} - 544 T^{5} + 29916 T^{6} - 544 p T^{7} + 1231 p^{2} T^{8} - 32 p^{3} T^{9} + 42 p^{4} T^{10} + p^{6} T^{12} )^{2} \) | |
| 23 | \( 1 - 60 T^{2} + 3354 T^{4} - 123436 T^{6} + 4153279 T^{8} - 110540568 T^{10} + 2811666956 T^{12} - 110540568 p^{2} T^{14} + 4153279 p^{4} T^{16} - 123436 p^{6} T^{18} + 3354 p^{8} T^{20} - 60 p^{10} T^{22} + p^{12} T^{24} \) | |
| 29 | \( ( 1 + 16 T + 200 T^{2} + 1728 T^{3} + 13363 T^{4} + 84496 T^{5} + 496976 T^{6} + 84496 p T^{7} + 13363 p^{2} T^{8} + 1728 p^{3} T^{9} + 200 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 31 | \( ( 1 - 8 T + 130 T^{2} - 824 T^{3} + 8247 T^{4} - 42224 T^{5} + 314764 T^{6} - 42224 p T^{7} + 8247 p^{2} T^{8} - 824 p^{3} T^{9} + 130 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 37 | \( ( 1 - 16 T + 200 T^{2} - 1680 T^{3} + 13131 T^{4} - 82528 T^{5} + 535824 T^{6} - 82528 p T^{7} + 13131 p^{2} T^{8} - 1680 p^{3} T^{9} + 200 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 41 | \( 1 - 264 T^{2} + 32344 T^{4} - 2436136 T^{6} + 127693075 T^{8} - 5269183920 T^{10} + 207449944752 T^{12} - 5269183920 p^{2} T^{14} + 127693075 p^{4} T^{16} - 2436136 p^{6} T^{18} + 32344 p^{8} T^{20} - 264 p^{10} T^{22} + p^{12} T^{24} \) | |
| 43 | \( 1 - 276 T^{2} + 36338 T^{4} - 3157924 T^{6} + 212530239 T^{8} - 11922137512 T^{10} + 561568704636 T^{12} - 11922137512 p^{2} T^{14} + 212530239 p^{4} T^{16} - 3157924 p^{6} T^{18} + 36338 p^{8} T^{20} - 276 p^{10} T^{22} + p^{12} T^{24} \) | |
| 47 | \( ( 1 + 90 T^{2} + 160 T^{3} + 6183 T^{4} + 13280 T^{5} + 328892 T^{6} + 13280 p T^{7} + 6183 p^{2} T^{8} + 160 p^{3} T^{9} + 90 p^{4} T^{10} + p^{6} T^{12} )^{2} \) | |
| 53 | \( ( 1 - 16 T + 342 T^{2} - 3952 T^{3} + 47367 T^{4} - 404032 T^{5} + 3397044 T^{6} - 404032 p T^{7} + 47367 p^{2} T^{8} - 3952 p^{3} T^{9} + 342 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 59 | \( ( 1 - 8 T + 210 T^{2} - 1624 T^{3} + 18415 T^{4} - 148976 T^{5} + 1121964 T^{6} - 148976 p T^{7} + 18415 p^{2} T^{8} - 1624 p^{3} T^{9} + 210 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 61 | \( 1 - 544 T^{2} + 142480 T^{4} - 23858720 T^{6} + 2849681139 T^{8} - 255863393088 T^{10} + 17709328352032 T^{12} - 255863393088 p^{2} T^{14} + 2849681139 p^{4} T^{16} - 23858720 p^{6} T^{18} + 142480 p^{8} T^{20} - 544 p^{10} T^{22} + p^{12} T^{24} \) | |
| 67 | \( 1 - 356 T^{2} + 55762 T^{4} - 4560756 T^{6} + 122644383 T^{8} + 13948776888 T^{10} - 1693776977860 T^{12} + 13948776888 p^{2} T^{14} + 122644383 p^{4} T^{16} - 4560756 p^{6} T^{18} + 55762 p^{8} T^{20} - 356 p^{10} T^{22} + p^{12} T^{24} \) | |
| 71 | \( 1 - 556 T^{2} + 151546 T^{4} - 27058972 T^{6} + 3535695487 T^{8} - 356059001528 T^{10} + 28330933942092 T^{12} - 356059001528 p^{2} T^{14} + 3535695487 p^{4} T^{16} - 27058972 p^{6} T^{18} + 151546 p^{8} T^{20} - 556 p^{10} T^{22} + p^{12} T^{24} \) | |
| 73 | \( 1 - 480 T^{2} + 105792 T^{4} - 14378720 T^{6} + 1390885347 T^{8} - 108814494400 T^{10} + 7928879090304 T^{12} - 108814494400 p^{2} T^{14} + 1390885347 p^{4} T^{16} - 14378720 p^{6} T^{18} + 105792 p^{8} T^{20} - 480 p^{10} T^{22} + p^{12} T^{24} \) | |
| 79 | \( 1 - 516 T^{2} + 134210 T^{4} - 23318900 T^{6} + 3036334383 T^{8} - 315920772616 T^{10} + 27245615350940 T^{12} - 315920772616 p^{2} T^{14} + 3036334383 p^{4} T^{16} - 23318900 p^{6} T^{18} + 134210 p^{8} T^{20} - 516 p^{10} T^{22} + p^{12} T^{24} \) | |
| 83 | \( ( 1 - 8 T + 290 T^{2} - 1848 T^{3} + 43207 T^{4} - 256016 T^{5} + 4402748 T^{6} - 256016 p T^{7} + 43207 p^{2} T^{8} - 1848 p^{3} T^{9} + 290 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 89 | \( 1 - 696 T^{2} + 241496 T^{4} - 54982424 T^{6} + 9125837523 T^{8} - 1161685318736 T^{10} + 116205471399600 T^{12} - 1161685318736 p^{2} T^{14} + 9125837523 p^{4} T^{16} - 54982424 p^{6} T^{18} + 241496 p^{8} T^{20} - 696 p^{10} T^{22} + p^{12} T^{24} \) | |
| 97 | \( 1 - 672 T^{2} + 218848 T^{4} - 46502304 T^{6} + 7344144003 T^{8} - 928722590784 T^{10} + 97980461927872 T^{12} - 928722590784 p^{2} T^{14} + 7344144003 p^{4} T^{16} - 46502304 p^{6} T^{18} + 218848 p^{8} T^{20} - 672 p^{10} T^{22} + p^{12} T^{24} \) | |
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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
−2.80572693948331517014110401689, −2.69797152942220852527873877754, −2.67521262362641180170451339780, −2.57443010512052862771545291996, −2.56011279356042462776664656845, −2.38085281326254528862695528430, −2.31341819715209911230591892701, −2.21479551985740209487045202492, −2.09407585733449697580226930526, −2.04142325801318312142705500857, −1.91037931692925213351977569495, −1.89047704649489461146404144516, −1.68641939867601166152092788370, −1.43517920561097840851495720472, −1.39262584169717754347452067075, −1.29142799974113542859289337937, −1.13624553834073537928284793577, −1.07593992070296351955054169667, −1.00709186437094583752640874075, −0.72858526609566145113387874499, −0.70123594313566234270088409998, −0.64464428242761397595903207352, −0.51389867933724660051545162159, −0.45686335943390899519367271316, −0.10926513226103249788606553191, 0.10926513226103249788606553191, 0.45686335943390899519367271316, 0.51389867933724660051545162159, 0.64464428242761397595903207352, 0.70123594313566234270088409998, 0.72858526609566145113387874499, 1.00709186437094583752640874075, 1.07593992070296351955054169667, 1.13624553834073537928284793577, 1.29142799974113542859289337937, 1.39262584169717754347452067075, 1.43517920561097840851495720472, 1.68641939867601166152092788370, 1.89047704649489461146404144516, 1.91037931692925213351977569495, 2.04142325801318312142705500857, 2.09407585733449697580226930526, 2.21479551985740209487045202492, 2.31341819715209911230591892701, 2.38085281326254528862695528430, 2.56011279356042462776664656845, 2.57443010512052862771545291996, 2.67521262362641180170451339780, 2.69797152942220852527873877754, 2.80572693948331517014110401689
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.