Dirichlet series
| L(s) = 1 | + 12·5-s − 12·9-s + 60·25-s + 18·29-s + 30·37-s − 6·41-s − 144·45-s − 39·49-s − 6·53-s + 36·61-s + 30·73-s + 75·81-s + 30·89-s + 42·97-s + 6·101-s + 108·109-s − 30·113-s − 66·121-s + 156·125-s + 127-s + 131-s + 137-s + 139-s + 216·145-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | + 5.36·5-s − 4·9-s + 12·25-s + 3.34·29-s + 4.93·37-s − 0.937·41-s − 21.4·45-s − 5.57·49-s − 0.824·53-s + 4.60·61-s + 3.51·73-s + 25/3·81-s + 3.17·89-s + 4.26·97-s + 0.597·101-s + 10.3·109-s − 2.82·113-s − 6·121-s + 13.9·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 17.9·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{60} \cdot 17^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{60} \cdot 17^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{60} \cdot 17^{24}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.62965\times 10^{22}\) |
| Root analytic conductor: | \(8.59334\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{60} \cdot 17^{24} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(267.7826814\) |
| \(L(\frac12)\) | \(\approx\) | \(267.7826814\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 \) | |
| good | 3 | \( 1 + 4 p T^{2} + 23 p T^{4} + 26 p^{2} T^{6} + 145 p T^{8} + 34 p T^{10} - 1369 T^{12} + 34 p^{3} T^{14} + 145 p^{5} T^{16} + 26 p^{8} T^{18} + 23 p^{9} T^{20} + 4 p^{11} T^{22} + p^{12} T^{24} \) |
| 5 | \( ( 1 - 6 T + 24 T^{2} - 78 T^{3} + 246 T^{4} - 642 T^{5} + 1527 T^{6} - 642 p T^{7} + 246 p^{2} T^{8} - 78 p^{3} T^{9} + 24 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 7 | \( 1 + 39 T^{2} + 99 p T^{4} + 7170 T^{6} + 45180 T^{8} + 165066 T^{10} + 551143 T^{12} + 165066 p^{2} T^{14} + 45180 p^{4} T^{16} + 7170 p^{6} T^{18} + 99 p^{9} T^{20} + 39 p^{10} T^{22} + p^{12} T^{24} \) | |
| 11 | \( 1 + 6 p T^{2} + 2451 T^{4} + 62741 T^{6} + 1214793 T^{8} + 18506385 T^{10} + 226515302 T^{12} + 18506385 p^{2} T^{14} + 1214793 p^{4} T^{16} + 62741 p^{6} T^{18} + 2451 p^{8} T^{20} + 6 p^{11} T^{22} + p^{12} T^{24} \) | |
| 13 | \( ( 1 + 36 T^{2} + T^{3} + 36 p T^{4} + p^{3} T^{6} )^{4} \) | |
| 19 | \( 1 + 60 T^{2} + 2418 T^{4} + 72382 T^{6} + 1945878 T^{8} + 44963910 T^{10} + 925697639 T^{12} + 44963910 p^{2} T^{14} + 1945878 p^{4} T^{16} + 72382 p^{6} T^{18} + 2418 p^{8} T^{20} + 60 p^{10} T^{22} + p^{12} T^{24} \) | |
| 23 | \( 1 + 87 T^{2} + 5715 T^{4} + 255952 T^{6} + 9503904 T^{8} + 281704086 T^{10} + 7140263783 T^{12} + 281704086 p^{2} T^{14} + 9503904 p^{4} T^{16} + 255952 p^{6} T^{18} + 5715 p^{8} T^{20} + 87 p^{10} T^{22} + p^{12} T^{24} \) | |
| 29 | \( ( 1 - 9 T + 141 T^{2} - 1094 T^{3} + 9540 T^{4} - 56838 T^{5} + 364885 T^{6} - 56838 p T^{7} + 9540 p^{2} T^{8} - 1094 p^{3} T^{9} + 141 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 31 | \( 1 + 171 T^{2} + 14637 T^{4} + 851086 T^{6} + 38379972 T^{8} + 1445416248 T^{10} + 47540110403 T^{12} + 1445416248 p^{2} T^{14} + 38379972 p^{4} T^{16} + 851086 p^{6} T^{18} + 14637 p^{8} T^{20} + 171 p^{10} T^{22} + p^{12} T^{24} \) | |
| 37 | \( ( 1 - 15 T + 273 T^{2} - 2660 T^{3} + 27612 T^{4} - 193230 T^{5} + 1397103 T^{6} - 193230 p T^{7} + 27612 p^{2} T^{8} - 2660 p^{3} T^{9} + 273 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 41 | \( ( 1 + 3 T + 159 T^{2} + 292 T^{3} + 12648 T^{4} + 19098 T^{5} + 649141 T^{6} + 19098 p T^{7} + 12648 p^{2} T^{8} + 292 p^{3} T^{9} + 159 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 43 | \( 1 + 174 T^{2} + 13581 T^{4} + 534858 T^{6} + 2733561 T^{8} - 955216290 T^{10} - 61202417345 T^{12} - 955216290 p^{2} T^{14} + 2733561 p^{4} T^{16} + 534858 p^{6} T^{18} + 13581 p^{8} T^{20} + 174 p^{10} T^{22} + p^{12} T^{24} \) | |
| 47 | \( 1 + 291 T^{2} + 43395 T^{4} + 4454862 T^{6} + 350705622 T^{8} + 22134896322 T^{10} + 1144853601583 T^{12} + 22134896322 p^{2} T^{14} + 350705622 p^{4} T^{16} + 4454862 p^{6} T^{18} + 43395 p^{8} T^{20} + 291 p^{10} T^{22} + p^{12} T^{24} \) | |
| 53 | \( ( 1 + 3 T + 138 T^{2} + 90 T^{3} + 10509 T^{4} + 7167 T^{5} + 669472 T^{6} + 7167 p T^{7} + 10509 p^{2} T^{8} + 90 p^{3} T^{9} + 138 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 59 | \( 1 + 324 T^{2} + 56385 T^{4} + 6787174 T^{6} + 10726545 p T^{8} + 48240529014 T^{10} + 3093040723031 T^{12} + 48240529014 p^{2} T^{14} + 10726545 p^{5} T^{16} + 6787174 p^{6} T^{18} + 56385 p^{8} T^{20} + 324 p^{10} T^{22} + p^{12} T^{24} \) | |
| 61 | \( ( 1 - 18 T + 369 T^{2} - 4406 T^{3} + 54069 T^{4} - 478554 T^{5} + 4309739 T^{6} - 478554 p T^{7} + 54069 p^{2} T^{8} - 4406 p^{3} T^{9} + 369 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 67 | \( 1 + 615 T^{2} + 181023 T^{4} + 33837166 T^{6} + 4484529462 T^{8} + 444696320994 T^{10} + 33865242402239 T^{12} + 444696320994 p^{2} T^{14} + 4484529462 p^{4} T^{16} + 33837166 p^{6} T^{18} + 181023 p^{8} T^{20} + 615 p^{10} T^{22} + p^{12} T^{24} \) | |
| 71 | \( 1 + 540 T^{2} + 147111 T^{4} + 26459192 T^{6} + 3481469163 T^{8} + 352259172150 T^{10} + 28076480971727 T^{12} + 352259172150 p^{2} T^{14} + 3481469163 p^{4} T^{16} + 26459192 p^{6} T^{18} + 147111 p^{8} T^{20} + 540 p^{10} T^{22} + p^{12} T^{24} \) | |
| 73 | \( ( 1 - 15 T + 243 T^{2} - 1594 T^{3} + 10578 T^{4} + 16974 T^{5} - 160933 T^{6} + 16974 p T^{7} + 10578 p^{2} T^{8} - 1594 p^{3} T^{9} + 243 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 79 | \( 1 + 309 T^{2} + 46731 T^{4} + 4551304 T^{6} + 349716933 T^{8} + 26057893635 T^{10} + 2036087910942 T^{12} + 26057893635 p^{2} T^{14} + 349716933 p^{4} T^{16} + 4551304 p^{6} T^{18} + 46731 p^{8} T^{20} + 309 p^{10} T^{22} + p^{12} T^{24} \) | |
| 83 | \( 1 + 330 T^{2} + 52965 T^{4} + 6142070 T^{6} + 607867506 T^{8} + 51265414146 T^{10} + 4095264799397 T^{12} + 51265414146 p^{2} T^{14} + 607867506 p^{4} T^{16} + 6142070 p^{6} T^{18} + 52965 p^{8} T^{20} + 330 p^{10} T^{22} + p^{12} T^{24} \) | |
| 89 | \( ( 1 - 15 T + 309 T^{2} - 3724 T^{3} + 47508 T^{4} - 440562 T^{5} + 56189 p T^{6} - 440562 p T^{7} + 47508 p^{2} T^{8} - 3724 p^{3} T^{9} + 309 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 97 | \( ( 1 - 21 T + 489 T^{2} - 7008 T^{3} + 104379 T^{4} - 1154247 T^{5} + 13050214 T^{6} - 1154247 p T^{7} + 104379 p^{2} T^{8} - 7008 p^{3} T^{9} + 489 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
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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
−2.21083005557806411747959822218, −2.06944129618394607474322355548, −2.01746648864230325068696212628, −1.99282372907852500502416541480, −1.91815996362921204365396445071, −1.87238789577199584527407176286, −1.84808599328811923035249088528, −1.79404884586889516087407158843, −1.75965788377860379642045704398, −1.61729658528538040322900515857, −1.59906616021012272765700646467, −1.57193742951505392705588468141, −1.34258210094607051661921323705, −1.17052640597129498211171689416, −1.07243455653697036758253999073, −0.929459758695261944430478914765, −0.844894807499597499992070230682, −0.812482392620028408769671201232, −0.69748665607045096857428534970, −0.64938889138478435268629837487, −0.63344757629914572730148750087, −0.55442662590848370411114842694, −0.44833293415779346090261712375, −0.28253684797331651262503326347, −0.19639568564417828656955332493, 0.19639568564417828656955332493, 0.28253684797331651262503326347, 0.44833293415779346090261712375, 0.55442662590848370411114842694, 0.63344757629914572730148750087, 0.64938889138478435268629837487, 0.69748665607045096857428534970, 0.812482392620028408769671201232, 0.844894807499597499992070230682, 0.929459758695261944430478914765, 1.07243455653697036758253999073, 1.17052640597129498211171689416, 1.34258210094607051661921323705, 1.57193742951505392705588468141, 1.59906616021012272765700646467, 1.61729658528538040322900515857, 1.75965788377860379642045704398, 1.79404884586889516087407158843, 1.84808599328811923035249088528, 1.87238789577199584527407176286, 1.91815996362921204365396445071, 1.99282372907852500502416541480, 2.01746648864230325068696212628, 2.06944129618394607474322355548, 2.21083005557806411747959822218
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.