Dirichlet series
| L(s) = 1 | − 8·3-s − 8·4-s − 16·5-s + 36·9-s + 64·12-s + 128·15-s + 36·16-s + 48·17-s + 128·20-s + 136·25-s − 120·27-s − 288·36-s + 16·41-s + 16·43-s − 576·45-s + 32·47-s − 288·48-s − 384·51-s + 32·59-s − 1.02e3·60-s − 120·64-s + 16·67-s − 384·68-s − 1.08e3·75-s − 48·79-s − 576·80-s + 330·81-s + ⋯ |
| L(s) = 1 | − 4.61·3-s − 4·4-s − 7.15·5-s + 12·9-s + 18.4·12-s + 33.0·15-s + 9·16-s + 11.6·17-s + 28.6·20-s + 27.1·25-s − 23.0·27-s − 48·36-s + 2.49·41-s + 2.43·43-s − 85.8·45-s + 4.66·47-s − 41.5·48-s − 53.7·51-s + 4.16·59-s − 132.·60-s − 15·64-s + 1.95·67-s − 46.5·68-s − 125.·75-s − 5.40·79-s − 64.3·80-s + 36.6·81-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 7^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.29870\times 10^{17}\) |
| Root analytic conductor: | \(3.42607\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 7^{32} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.4678710322\) |
| \(L(\frac12)\) | \(\approx\) | \(0.4678710322\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T^{2} )^{8} \) |
| 3 | \( 1 + 8 T + 28 T^{2} + 56 T^{3} + 70 T^{4} + 56 T^{5} + 28 T^{6} + 8 T^{7} + 2 T^{8} + 8 p T^{9} + 28 p^{2} T^{10} + 56 p^{3} T^{11} + 70 p^{4} T^{12} + 56 p^{5} T^{13} + 28 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 5 | \( ( 1 + T )^{16} \) | |
| 7 | \( 1 \) | |
| good | 11 | \( 1 - 96 T^{2} + 4292 T^{4} - 120480 T^{6} + 2453256 T^{8} - 40385632 T^{10} + 583520268 T^{12} - 7638851744 T^{14} + 89425732174 T^{16} - 7638851744 p^{2} T^{18} + 583520268 p^{4} T^{20} - 40385632 p^{6} T^{22} + 2453256 p^{8} T^{24} - 120480 p^{10} T^{26} + 4292 p^{12} T^{28} - 96 p^{14} T^{30} + p^{16} T^{32} \) |
| 13 | \( 1 - 120 T^{2} + 7392 T^{4} - 307304 T^{6} + 9589500 T^{8} - 237192120 T^{10} + 4797255456 T^{12} - 80787529192 T^{14} + 1143723641414 T^{16} - 80787529192 p^{2} T^{18} + 4797255456 p^{4} T^{20} - 237192120 p^{6} T^{22} + 9589500 p^{8} T^{24} - 307304 p^{10} T^{26} + 7392 p^{12} T^{28} - 120 p^{14} T^{30} + p^{16} T^{32} \) | |
| 17 | \( ( 1 - 24 T + 328 T^{2} - 3160 T^{3} + 24080 T^{4} - 152664 T^{5} + 837192 T^{6} - 4052504 T^{7} + 17610626 T^{8} - 4052504 p T^{9} + 837192 p^{2} T^{10} - 152664 p^{3} T^{11} + 24080 p^{4} T^{12} - 3160 p^{5} T^{13} + 328 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 19 | \( 1 - 136 T^{2} + 9628 T^{4} - 466872 T^{6} + 17249224 T^{8} - 27174872 p T^{10} + 13084084116 T^{12} - 290595451192 T^{14} + 5798268762702 T^{16} - 290595451192 p^{2} T^{18} + 13084084116 p^{4} T^{20} - 27174872 p^{7} T^{22} + 17249224 p^{8} T^{24} - 466872 p^{10} T^{26} + 9628 p^{12} T^{28} - 136 p^{14} T^{30} + p^{16} T^{32} \) | |
| 23 | \( 1 - 120 T^{2} + 8416 T^{4} - 437288 T^{6} + 18476732 T^{8} - 662624952 T^{10} + 20672216352 T^{12} - 568229031336 T^{14} + 13863549940486 T^{16} - 568229031336 p^{2} T^{18} + 20672216352 p^{4} T^{20} - 662624952 p^{6} T^{22} + 18476732 p^{8} T^{24} - 437288 p^{10} T^{26} + 8416 p^{12} T^{28} - 120 p^{14} T^{30} + p^{16} T^{32} \) | |
| 29 | \( 1 - 280 T^{2} + 36956 T^{4} - 3049768 T^{6} + 176284488 T^{8} - 7622160664 T^{10} + 261486844116 T^{12} - 7736075388456 T^{14} + 221103264309838 T^{16} - 7736075388456 p^{2} T^{18} + 261486844116 p^{4} T^{20} - 7622160664 p^{6} T^{22} + 176284488 p^{8} T^{24} - 3049768 p^{10} T^{26} + 36956 p^{12} T^{28} - 280 p^{14} T^{30} + p^{16} T^{32} \) | |
| 31 | \( 1 - 216 T^{2} + 23772 T^{4} - 1801640 T^{6} + 107110344 T^{8} - 5323330776 T^{10} + 227557158036 T^{12} - 8491047861160 T^{14} + 279568671077774 T^{16} - 8491047861160 p^{2} T^{18} + 227557158036 p^{4} T^{20} - 5323330776 p^{6} T^{22} + 107110344 p^{8} T^{24} - 1801640 p^{10} T^{26} + 23772 p^{12} T^{28} - 216 p^{14} T^{30} + p^{16} T^{32} \) | |
| 37 | \( ( 1 + 84 T^{2} + 96 T^{3} + 4758 T^{4} + 18768 T^{5} + 164900 T^{6} + 1237136 T^{7} + 5757282 T^{8} + 1237136 p T^{9} + 164900 p^{2} T^{10} + 18768 p^{3} T^{11} + 4758 p^{4} T^{12} + 96 p^{5} T^{13} + 84 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 41 | \( ( 1 - 8 T + 140 T^{2} - 760 T^{3} + 8134 T^{4} - 31640 T^{5} + 283532 T^{6} - 664168 T^{7} + 8644802 T^{8} - 664168 p T^{9} + 283532 p^{2} T^{10} - 31640 p^{3} T^{11} + 8134 p^{4} T^{12} - 760 p^{5} T^{13} + 140 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 43 | \( ( 1 - 8 T + 216 T^{2} - 1336 T^{3} + 18930 T^{4} - 91624 T^{5} + 941992 T^{6} - 3893656 T^{7} + 38489570 T^{8} - 3893656 p T^{9} + 941992 p^{2} T^{10} - 91624 p^{3} T^{11} + 18930 p^{4} T^{12} - 1336 p^{5} T^{13} + 216 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 47 | \( ( 1 - 16 T + 308 T^{2} - 3456 T^{3} + 41878 T^{4} - 380384 T^{5} + 3526276 T^{6} - 26392816 T^{7} + 198920994 T^{8} - 26392816 p T^{9} + 3526276 p^{2} T^{10} - 380384 p^{3} T^{11} + 41878 p^{4} T^{12} - 3456 p^{5} T^{13} + 308 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 53 | \( 1 - 464 T^{2} + 103672 T^{4} - 15090416 T^{6} + 1639444188 T^{8} - 144167035088 T^{10} + 10766240671304 T^{12} - 698236845794160 T^{14} + 39574123695783302 T^{16} - 698236845794160 p^{2} T^{18} + 10766240671304 p^{4} T^{20} - 144167035088 p^{6} T^{22} + 1639444188 p^{8} T^{24} - 15090416 p^{10} T^{26} + 103672 p^{12} T^{28} - 464 p^{14} T^{30} + p^{16} T^{32} \) | |
| 59 | \( ( 1 - 16 T + 416 T^{2} - 4912 T^{3} + 74498 T^{4} - 706464 T^{5} + 7978496 T^{6} - 62656160 T^{7} + 569371650 T^{8} - 62656160 p T^{9} + 7978496 p^{2} T^{10} - 706464 p^{3} T^{11} + 74498 p^{4} T^{12} - 4912 p^{5} T^{13} + 416 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 61 | \( 1 - 256 T^{2} + 40248 T^{4} - 4546304 T^{6} + 413213660 T^{8} - 31789913344 T^{10} + 2157545663240 T^{12} - 136369557658368 T^{14} + 8328401549152262 T^{16} - 136369557658368 p^{2} T^{18} + 2157545663240 p^{4} T^{20} - 31789913344 p^{6} T^{22} + 413213660 p^{8} T^{24} - 4546304 p^{10} T^{26} + 40248 p^{12} T^{28} - 256 p^{14} T^{30} + p^{16} T^{32} \) | |
| 67 | \( ( 1 - 8 T + 236 T^{2} - 1960 T^{3} + 34678 T^{4} - 247256 T^{5} + 3447836 T^{6} - 22024760 T^{7} + 261423522 T^{8} - 22024760 p T^{9} + 3447836 p^{2} T^{10} - 247256 p^{3} T^{11} + 34678 p^{4} T^{12} - 1960 p^{5} T^{13} + 236 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 71 | \( 1 - 384 T^{2} + 89208 T^{4} - 15037568 T^{6} + 2022630044 T^{8} - 227346411904 T^{10} + 22005424210504 T^{12} - 1867790290274432 T^{14} + 140615918254013638 T^{16} - 1867790290274432 p^{2} T^{18} + 22005424210504 p^{4} T^{20} - 227346411904 p^{6} T^{22} + 2022630044 p^{8} T^{24} - 15037568 p^{10} T^{26} + 89208 p^{12} T^{28} - 384 p^{14} T^{30} + p^{16} T^{32} \) | |
| 73 | \( 1 - 408 T^{2} + 103836 T^{4} - 18378408 T^{6} + 2577950792 T^{8} - 295004324952 T^{10} + 29031392627796 T^{12} - 2492858264234856 T^{14} + 192358487967014926 T^{16} - 2492858264234856 p^{2} T^{18} + 29031392627796 p^{4} T^{20} - 295004324952 p^{6} T^{22} + 2577950792 p^{8} T^{24} - 18378408 p^{10} T^{26} + 103836 p^{12} T^{28} - 408 p^{14} T^{30} + p^{16} T^{32} \) | |
| 79 | \( ( 1 + 24 T + 656 T^{2} + 11256 T^{3} + 185532 T^{4} + 2408280 T^{5} + 29518320 T^{6} + 302426232 T^{7} + 2917100806 T^{8} + 302426232 p T^{9} + 29518320 p^{2} T^{10} + 2408280 p^{3} T^{11} + 185532 p^{4} T^{12} + 11256 p^{5} T^{13} + 656 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 83 | \( ( 1 - 24 T + 412 T^{2} - 3608 T^{3} + 28102 T^{4} - 159960 T^{5} + 2919132 T^{6} - 38138264 T^{7} + 469201730 T^{8} - 38138264 p T^{9} + 2919132 p^{2} T^{10} - 159960 p^{3} T^{11} + 28102 p^{4} T^{12} - 3608 p^{5} T^{13} + 412 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 89 | \( ( 1 - 8 T + 572 T^{2} - 3960 T^{3} + 148774 T^{4} - 885112 T^{5} + 23442748 T^{6} - 118982408 T^{7} + 2496928386 T^{8} - 118982408 p T^{9} + 23442748 p^{2} T^{10} - 885112 p^{3} T^{11} + 148774 p^{4} T^{12} - 3960 p^{5} T^{13} + 572 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 97 | \( 1 - 536 T^{2} + 122460 T^{4} - 13005352 T^{6} + 23401288 T^{8} + 159493931176 T^{10} - 15222534310380 T^{12} - 283840369941352 T^{14} + 136563954276810894 T^{16} - 283840369941352 p^{2} T^{18} - 15222534310380 p^{4} T^{20} + 159493931176 p^{6} T^{22} + 23401288 p^{8} T^{24} - 13005352 p^{10} T^{26} + 122460 p^{12} T^{28} - 536 p^{14} T^{30} + p^{16} T^{32} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.47504885498831177970387229629, −2.41809222094656450061682683719, −2.29834200837101254130252007735, −2.06437520836450601608665793124, −1.98595340164034972120638848574, −1.92786832399819344420416628726, −1.83922147768863321234121275955, −1.68732874308336656437607087195, −1.61843790281807570079962784459, −1.57099076177057160170458766328, −1.29085556857851338640145725127, −1.13302117020289484286722817818, −1.10485354929085499772141307733, −1.04654472236607081296115102344, −1.02395586436805386858193094452, −0.925846228721370166294544132123, −0.922797169524118874712396185412, −0.803167768224590319799090856839, −0.73638105684128272344337836360, −0.67816219584290587372632438172, −0.64529370132985968692820939006, −0.51262534123031199535966721987, −0.49776688382893755811103114975, −0.42333522564174161867751862787, −0.12181769599622481448303920096, 0.12181769599622481448303920096, 0.42333522564174161867751862787, 0.49776688382893755811103114975, 0.51262534123031199535966721987, 0.64529370132985968692820939006, 0.67816219584290587372632438172, 0.73638105684128272344337836360, 0.803167768224590319799090856839, 0.922797169524118874712396185412, 0.925846228721370166294544132123, 1.02395586436805386858193094452, 1.04654472236607081296115102344, 1.10485354929085499772141307733, 1.13302117020289484286722817818, 1.29085556857851338640145725127, 1.57099076177057160170458766328, 1.61843790281807570079962784459, 1.68732874308336656437607087195, 1.83922147768863321234121275955, 1.92786832399819344420416628726, 1.98595340164034972120638848574, 2.06437520836450601608665793124, 2.29834200837101254130252007735, 2.41809222094656450061682683719, 2.47504885498831177970387229629
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.