Dirichlet series
| L(s) = 1 | + 16·2-s + 136·4-s − 16·5-s + 816·8-s − 256·10-s − 20·11-s + 3.87e3·16-s − 8·19-s − 2.17e3·20-s − 320·22-s + 136·25-s + 1.55e4·32-s + 32·37-s − 128·38-s − 1.30e4·40-s − 8·41-s − 2.72e3·44-s + 12·49-s + 2.17e3·50-s + 8·53-s + 320·55-s + 5.42e4·64-s + 4·67-s + 16·73-s + 512·74-s − 1.08e3·76-s − 6.20e4·80-s + ⋯ |
| L(s) = 1 | + 11.3·2-s + 68·4-s − 7.15·5-s + 288.·8-s − 80.9·10-s − 6.03·11-s + 969·16-s − 1.83·19-s − 486.·20-s − 68.2·22-s + 27.1·25-s + 2.74e3·32-s + 5.26·37-s − 20.7·38-s − 2.06e3·40-s − 1.24·41-s − 410.·44-s + 12/7·49-s + 307.·50-s + 1.09·53-s + 43.1·55-s + 6.78e3·64-s + 0.488·67-s + 1.87·73-s + 59.5·74-s − 124.·76-s − 6.93e3·80-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 5^{16} \cdot 67^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 5^{16} \cdot 67^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{16} \cdot 3^{32} \cdot 5^{16} \cdot 67^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(8.34662\times 10^{26}\) |
| Root analytic conductor: | \(6.93900\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{16} \cdot 3^{32} \cdot 5^{16} \cdot 67^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(23931.20309\) |
| \(L(\frac12)\) | \(\approx\) | \(23931.20309\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T )^{16} \) |
| 3 | \( 1 \) | |
| 5 | \( ( 1 + T )^{16} \) | |
| 67 | \( 1 - 4 T - 144 T^{2} + 1052 T^{3} + 6556 T^{4} - 111204 T^{5} + 538512 T^{6} + 3644572 T^{7} - 74118810 T^{8} + 3644572 p T^{9} + 538512 p^{2} T^{10} - 111204 p^{3} T^{11} + 6556 p^{4} T^{12} + 1052 p^{5} T^{13} - 144 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| good | 7 | \( 1 - 12 T^{2} + 65 T^{4} - 64 T^{6} + 10 p T^{8} + 1000 T^{10} + 113791 T^{12} - 1570844 T^{14} + 15564210 T^{16} - 1570844 p^{2} T^{18} + 113791 p^{4} T^{20} + 1000 p^{6} T^{22} + 10 p^{9} T^{24} - 64 p^{10} T^{26} + 65 p^{12} T^{28} - 12 p^{14} T^{30} + p^{16} T^{32} \) |
| 11 | \( ( 1 + 10 T + 89 T^{2} + 566 T^{3} + 3224 T^{4} + 15446 T^{5} + 67469 T^{6} + 258178 T^{7} + 910522 T^{8} + 258178 p T^{9} + 67469 p^{2} T^{10} + 15446 p^{3} T^{11} + 3224 p^{4} T^{12} + 566 p^{5} T^{13} + 89 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 13 | \( 1 - 100 T^{2} + 5164 T^{4} - 182812 T^{6} + 4978036 T^{8} - 110574964 T^{10} + 2069794196 T^{12} - 33262092620 T^{14} + 463566233430 T^{16} - 33262092620 p^{2} T^{18} + 2069794196 p^{4} T^{20} - 110574964 p^{6} T^{22} + 4978036 p^{8} T^{24} - 182812 p^{10} T^{26} + 5164 p^{12} T^{28} - 100 p^{14} T^{30} + p^{16} T^{32} \) | |
| 17 | \( 1 - 72 T^{2} + 3388 T^{4} - 119208 T^{6} + 3472552 T^{8} - 87055608 T^{10} + 113187508 p T^{12} - 38031267608 T^{14} + 679448411470 T^{16} - 38031267608 p^{2} T^{18} + 113187508 p^{5} T^{20} - 87055608 p^{6} T^{22} + 3472552 p^{8} T^{24} - 119208 p^{10} T^{26} + 3388 p^{12} T^{28} - 72 p^{14} T^{30} + p^{16} T^{32} \) | |
| 19 | \( ( 1 + 4 T + 52 T^{2} + 132 T^{3} + 1720 T^{4} + 5684 T^{5} + 50044 T^{6} + 126740 T^{7} + 49850 p T^{8} + 126740 p T^{9} + 50044 p^{2} T^{10} + 5684 p^{3} T^{11} + 1720 p^{4} T^{12} + 132 p^{5} T^{13} + 52 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 23 | \( 1 - 236 T^{2} + 26820 T^{4} - 1953524 T^{6} + 102662900 T^{8} - 4171202524 T^{10} + 137809451964 T^{12} - 3858496068548 T^{14} + 94388796186262 T^{16} - 3858496068548 p^{2} T^{18} + 137809451964 p^{4} T^{20} - 4171202524 p^{6} T^{22} + 102662900 p^{8} T^{24} - 1953524 p^{10} T^{26} + 26820 p^{12} T^{28} - 236 p^{14} T^{30} + p^{16} T^{32} \) | |
| 29 | \( 1 - 4 p T^{2} + 5532 T^{4} - 153676 T^{6} + 3478164 T^{8} - 93695876 T^{10} + 2804533284 T^{12} - 62769871100 T^{14} + 1298249772566 T^{16} - 62769871100 p^{2} T^{18} + 2804533284 p^{4} T^{20} - 93695876 p^{6} T^{22} + 3478164 p^{8} T^{24} - 153676 p^{10} T^{26} + 5532 p^{12} T^{28} - 4 p^{15} T^{30} + p^{16} T^{32} \) | |
| 31 | \( 1 - 356 T^{2} + 61544 T^{4} - 6865052 T^{6} + 554409616 T^{8} - 34496199012 T^{10} + 1717973499704 T^{12} - 70142618146140 T^{14} + 2379955389522206 T^{16} - 70142618146140 p^{2} T^{18} + 1717973499704 p^{4} T^{20} - 34496199012 p^{6} T^{22} + 554409616 p^{8} T^{24} - 6865052 p^{10} T^{26} + 61544 p^{12} T^{28} - 356 p^{14} T^{30} + p^{16} T^{32} \) | |
| 37 | \( ( 1 - 16 T + 269 T^{2} - 2480 T^{3} + 24724 T^{4} - 168876 T^{5} + 1334745 T^{6} - 7746124 T^{7} + 54540506 T^{8} - 7746124 p T^{9} + 1334745 p^{2} T^{10} - 168876 p^{3} T^{11} + 24724 p^{4} T^{12} - 2480 p^{5} T^{13} + 269 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 41 | \( ( 1 + 4 T + 88 T^{2} + 192 T^{3} + 3730 T^{4} + 12008 T^{5} + 169712 T^{6} + 787124 T^{7} + 7094090 T^{8} + 787124 p T^{9} + 169712 p^{2} T^{10} + 12008 p^{3} T^{11} + 3730 p^{4} T^{12} + 192 p^{5} T^{13} + 88 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 43 | \( 1 - 88 T^{2} + 9648 T^{4} - 590152 T^{6} + 36665500 T^{8} - 1823293528 T^{10} + 84892264016 T^{12} - 3881157187976 T^{14} + 160891788941702 T^{16} - 3881157187976 p^{2} T^{18} + 84892264016 p^{4} T^{20} - 1823293528 p^{6} T^{22} + 36665500 p^{8} T^{24} - 590152 p^{10} T^{26} + 9648 p^{12} T^{28} - 88 p^{14} T^{30} + p^{16} T^{32} \) | |
| 47 | \( 1 - 236 T^{2} + 31140 T^{4} - 3102004 T^{6} + 253966836 T^{8} - 17838006620 T^{10} + 1103973200988 T^{12} - 60912719078276 T^{14} + 3016913541521174 T^{16} - 60912719078276 p^{2} T^{18} + 1103973200988 p^{4} T^{20} - 17838006620 p^{6} T^{22} + 253966836 p^{8} T^{24} - 3102004 p^{10} T^{26} + 31140 p^{12} T^{28} - 236 p^{14} T^{30} + p^{16} T^{32} \) | |
| 53 | \( ( 1 - 4 T + 172 T^{2} - 124 T^{3} + 13704 T^{4} + 18028 T^{5} + 1035940 T^{6} + 527028 T^{7} + 68170350 T^{8} + 527028 p T^{9} + 1035940 p^{2} T^{10} + 18028 p^{3} T^{11} + 13704 p^{4} T^{12} - 124 p^{5} T^{13} + 172 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 59 | \( 1 - 584 T^{2} + 167280 T^{4} - 31375000 T^{6} + 4345877148 T^{8} - 474929551176 T^{10} + 42593872173456 T^{12} - 3206861640064088 T^{14} + 204993910642000518 T^{16} - 3206861640064088 p^{2} T^{18} + 42593872173456 p^{4} T^{20} - 474929551176 p^{6} T^{22} + 4345877148 p^{8} T^{24} - 31375000 p^{10} T^{26} + 167280 p^{12} T^{28} - 584 p^{14} T^{30} + p^{16} T^{32} \) | |
| 61 | \( 1 - 164 T^{2} + 23637 T^{4} - 2264768 T^{6} + 219737070 T^{8} - 17352386456 T^{10} + 1361333769007 T^{12} - 88851384720820 T^{14} + 5835679164863962 T^{16} - 88851384720820 p^{2} T^{18} + 1361333769007 p^{4} T^{20} - 17352386456 p^{6} T^{22} + 219737070 p^{8} T^{24} - 2264768 p^{10} T^{26} + 23637 p^{12} T^{28} - 164 p^{14} T^{30} + p^{16} T^{32} \) | |
| 71 | \( 1 - 436 T^{2} + 103865 T^{4} - 17205272 T^{6} + 2228671174 T^{8} - 240033414624 T^{10} + 22427523978183 T^{12} - 1860695785363924 T^{14} + 139019003009222450 T^{16} - 1860695785363924 p^{2} T^{18} + 22427523978183 p^{4} T^{20} - 240033414624 p^{6} T^{22} + 2228671174 p^{8} T^{24} - 17205272 p^{10} T^{26} + 103865 p^{12} T^{28} - 436 p^{14} T^{30} + p^{16} T^{32} \) | |
| 73 | \( ( 1 - 8 T + 448 T^{2} - 2888 T^{3} + 94100 T^{4} - 507992 T^{5} + 12229600 T^{6} - 55637848 T^{7} + 1073875926 T^{8} - 55637848 p T^{9} + 12229600 p^{2} T^{10} - 507992 p^{3} T^{11} + 94100 p^{4} T^{12} - 2888 p^{5} T^{13} + 448 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 79 | \( 1 - 692 T^{2} + 251368 T^{4} - 62358860 T^{6} + 11696423312 T^{8} - 1746760099060 T^{10} + 213912626620344 T^{12} - 21851789495031500 T^{14} + 1878932976025351710 T^{16} - 21851789495031500 p^{2} T^{18} + 213912626620344 p^{4} T^{20} - 1746760099060 p^{6} T^{22} + 11696423312 p^{8} T^{24} - 62358860 p^{10} T^{26} + 251368 p^{12} T^{28} - 692 p^{14} T^{30} + p^{16} T^{32} \) | |
| 83 | \( 1 - 472 T^{2} + 119705 T^{4} - 21970144 T^{6} + 3253765858 T^{8} - 410263950336 T^{10} + 45393778890011 T^{12} - 4461334566964584 T^{14} + 391452390769526978 T^{16} - 4461334566964584 p^{2} T^{18} + 45393778890011 p^{4} T^{20} - 410263950336 p^{6} T^{22} + 3253765858 p^{8} T^{24} - 21970144 p^{10} T^{26} + 119705 p^{12} T^{28} - 472 p^{14} T^{30} + p^{16} T^{32} \) | |
| 89 | \( 1 - 496 T^{2} + 136613 T^{4} - 27528624 T^{6} + 4522731274 T^{8} - 631608552016 T^{10} + 76556075269003 T^{12} - 8155835267046416 T^{14} + 770110258891235850 T^{16} - 8155835267046416 p^{2} T^{18} + 76556075269003 p^{4} T^{20} - 631608552016 p^{6} T^{22} + 4522731274 p^{8} T^{24} - 27528624 p^{10} T^{26} + 136613 p^{12} T^{28} - 496 p^{14} T^{30} + p^{16} T^{32} \) | |
| 97 | \( 1 - 832 T^{2} + 338597 T^{4} - 90113976 T^{6} + 17744840082 T^{8} - 2779884709240 T^{10} + 364561254235739 T^{12} - 41610271866017936 T^{14} + 4246834307168352730 T^{16} - 41610271866017936 p^{2} T^{18} + 364561254235739 p^{4} T^{20} - 2779884709240 p^{6} T^{22} + 17744840082 p^{8} T^{24} - 90113976 p^{10} T^{26} + 338597 p^{12} T^{28} - 832 p^{14} T^{30} + p^{16} T^{32} \) | |
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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
−2.12436144780281080625246270365, −2.11539187446441541619656759542, −2.02654397014431066751551247170, −1.97214590049130125989603107398, −1.83619809114205224329121672555, −1.78682535561263670270351115634, −1.70765980999313689453057093158, −1.61945297815794657240155188355, −1.51587961478161171585036942022, −1.36802585401607003355168178834, −1.29815167141310847809109664658, −1.26772222788650241996225724607, −1.10758983000988294089209944017, −1.05924346065079632199741611877, −1.02281591837542674848149296910, −1.00315769465277616228925581640, −0.905843417216140484322585595997, −0.78345596910777961087033422894, −0.57624313185614307491971381242, −0.53957125186804389033477568960, −0.44059612674100718735249809430, −0.41674848810254780175203684849, −0.34061180276320126310089368173, −0.25010096201736592292993700852, −0.14851205362908860547421783310, 0.14851205362908860547421783310, 0.25010096201736592292993700852, 0.34061180276320126310089368173, 0.41674848810254780175203684849, 0.44059612674100718735249809430, 0.53957125186804389033477568960, 0.57624313185614307491971381242, 0.78345596910777961087033422894, 0.905843417216140484322585595997, 1.00315769465277616228925581640, 1.02281591837542674848149296910, 1.05924346065079632199741611877, 1.10758983000988294089209944017, 1.26772222788650241996225724607, 1.29815167141310847809109664658, 1.36802585401607003355168178834, 1.51587961478161171585036942022, 1.61945297815794657240155188355, 1.70765980999313689453057093158, 1.78682535561263670270351115634, 1.83619809114205224329121672555, 1.97214590049130125989603107398, 2.02654397014431066751551247170, 2.11539187446441541619656759542, 2.12436144780281080625246270365
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.