Dirichlet series
| L(s) = 1 | − 1.49e3·9-s + 3.05e4·19-s + 7.00e4·31-s + 7.38e5·49-s − 2.71e5·61-s − 1.41e6·79-s + 5.13e5·81-s + 9.92e6·109-s + 1.60e7·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5.75e7·169-s − 4.55e7·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 2.04·9-s + 4.45·19-s + 2.35·31-s + 6.27·49-s − 1.19·61-s − 2.87·79-s + 0.967·81-s + 7.66·109-s + 9.05·121-s + 11.9·169-s − 9.11·171-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s+3)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{32} \cdot 3^{16} \cdot 5^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.64983\times 10^{29}\) |
| Root analytic conductor: | \(8.30760\) |
| Motivic weight: | \(6\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{32} \cdot 3^{16} \cdot 5^{32} ,\ ( \ : [3]^{16} ),\ 1 )\) |
Particular Values
| \(L(\frac{7}{2})\) | \(\approx\) | \(90.37236249\) |
| \(L(\frac12)\) | \(\approx\) | \(90.37236249\) |
| \(L(4)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1492 T^{2} + 190232 p^{2} T^{4} + 12721564 p^{4} T^{6} + 13637630 p^{10} T^{8} + 12721564 p^{16} T^{10} + 190232 p^{26} T^{12} + 1492 p^{36} T^{14} + p^{48} T^{16} \) | |
| 5 | \( 1 \) | |
| good | 7 | \( ( 1 - 369316 T^{2} + 95995178200 T^{4} - 17109232557961804 T^{6} + \)\(23\!\cdots\!54\)\( T^{8} - 17109232557961804 p^{12} T^{10} + 95995178200 p^{24} T^{12} - 369316 p^{36} T^{14} + p^{48} T^{16} )^{2} \) |
| 11 | \( ( 1 - 8019728 T^{2} + 33285766478428 T^{4} - 92444356768450321136 T^{6} + \)\(18\!\cdots\!70\)\( T^{8} - 92444356768450321136 p^{12} T^{10} + 33285766478428 p^{24} T^{12} - 8019728 p^{36} T^{14} + p^{48} T^{16} )^{2} \) | |
| 13 | \( ( 1 - 28750336 T^{2} + 387753606918460 T^{4} - \)\(32\!\cdots\!64\)\( T^{6} + \)\(18\!\cdots\!34\)\( T^{8} - \)\(32\!\cdots\!64\)\( p^{12} T^{10} + 387753606918460 p^{24} T^{12} - 28750336 p^{36} T^{14} + p^{48} T^{16} )^{2} \) | |
| 17 | \( ( 1 + 31278392 T^{2} + 172821670381468 T^{4} - \)\(12\!\cdots\!96\)\( T^{6} - \)\(59\!\cdots\!30\)\( T^{8} - \)\(12\!\cdots\!96\)\( p^{12} T^{10} + 172821670381468 p^{24} T^{12} + 31278392 p^{36} T^{14} + p^{48} T^{16} )^{2} \) | |
| 19 | \( ( 1 - 7636 T + 109941760 T^{2} - 622372357564 T^{3} + 7054529871207934 T^{4} - 622372357564 p^{6} T^{5} + 109941760 p^{12} T^{6} - 7636 p^{18} T^{7} + p^{24} T^{8} )^{4} \) | |
| 23 | \( ( 1 + 468867572 T^{2} + 121906152183901528 T^{4} + \)\(25\!\cdots\!64\)\( T^{6} + \)\(43\!\cdots\!70\)\( T^{8} + \)\(25\!\cdots\!64\)\( p^{12} T^{10} + 121906152183901528 p^{24} T^{12} + 468867572 p^{36} T^{14} + p^{48} T^{16} )^{2} \) | |
| 29 | \( ( 1 - 2475312128 T^{2} + 2545049575721746108 T^{4} - \)\(14\!\cdots\!16\)\( T^{6} + \)\(77\!\cdots\!70\)\( T^{8} - \)\(14\!\cdots\!16\)\( p^{12} T^{10} + 2545049575721746108 p^{24} T^{12} - 2475312128 p^{36} T^{14} + p^{48} T^{16} )^{2} \) | |
| 31 | \( ( 1 - 17516 T + 1295746120 T^{2} + 8029412140156 T^{3} + 767980698108536974 T^{4} + 8029412140156 p^{6} T^{5} + 1295746120 p^{12} T^{6} - 17516 p^{18} T^{7} + p^{24} T^{8} )^{4} \) | |
| 37 | \( ( 1 - 12756697408 T^{2} + 78999718552608402748 T^{4} - \)\(32\!\cdots\!76\)\( T^{6} + \)\(94\!\cdots\!70\)\( T^{8} - \)\(32\!\cdots\!76\)\( p^{12} T^{10} + 78999718552608402748 p^{24} T^{12} - 12756697408 p^{36} T^{14} + p^{48} T^{16} )^{2} \) | |
| 41 | \( ( 1 - 13583881088 T^{2} + \)\(11\!\cdots\!28\)\( T^{4} - \)\(70\!\cdots\!76\)\( T^{6} + \)\(36\!\cdots\!70\)\( T^{8} - \)\(70\!\cdots\!76\)\( p^{12} T^{10} + \)\(11\!\cdots\!28\)\( p^{24} T^{12} - 13583881088 p^{36} T^{14} + p^{48} T^{16} )^{2} \) | |
| 43 | \( ( 1 - 13680907276 T^{2} + \)\(13\!\cdots\!20\)\( T^{4} - \)\(11\!\cdots\!64\)\( T^{6} + \)\(84\!\cdots\!34\)\( T^{8} - \)\(11\!\cdots\!64\)\( p^{12} T^{10} + \)\(13\!\cdots\!20\)\( p^{24} T^{12} - 13680907276 p^{36} T^{14} + p^{48} T^{16} )^{2} \) | |
| 47 | \( ( 1 + 50505761492 T^{2} + \)\(11\!\cdots\!88\)\( T^{4} + \)\(18\!\cdots\!84\)\( T^{6} + \)\(21\!\cdots\!70\)\( T^{8} + \)\(18\!\cdots\!84\)\( p^{12} T^{10} + \)\(11\!\cdots\!88\)\( p^{24} T^{12} + 50505761492 p^{36} T^{14} + p^{48} T^{16} )^{2} \) | |
| 53 | \( ( 1 + 79723023992 T^{2} + \)\(41\!\cdots\!88\)\( T^{4} + \)\(14\!\cdots\!84\)\( T^{6} + \)\(37\!\cdots\!70\)\( T^{8} + \)\(14\!\cdots\!84\)\( p^{12} T^{10} + \)\(41\!\cdots\!88\)\( p^{24} T^{12} + 79723023992 p^{36} T^{14} + p^{48} T^{16} )^{2} \) | |
| 59 | \( ( 1 - 71542760768 T^{2} + \)\(14\!\cdots\!08\)\( T^{4} + \)\(26\!\cdots\!24\)\( T^{6} - \)\(19\!\cdots\!30\)\( T^{8} + \)\(26\!\cdots\!24\)\( p^{12} T^{10} + \)\(14\!\cdots\!08\)\( p^{24} T^{12} - 71542760768 p^{36} T^{14} + p^{48} T^{16} )^{2} \) | |
| 61 | \( ( 1 + 67804 T + 143593798600 T^{2} + 11191577001643636 T^{3} + \)\(99\!\cdots\!34\)\( T^{4} + 11191577001643636 p^{6} T^{5} + 143593798600 p^{12} T^{6} + 67804 p^{18} T^{7} + p^{24} T^{8} )^{4} \) | |
| 67 | \( ( 1 - 512947080076 T^{2} + \)\(12\!\cdots\!60\)\( T^{4} - \)\(17\!\cdots\!44\)\( T^{6} + \)\(18\!\cdots\!74\)\( T^{8} - \)\(17\!\cdots\!44\)\( p^{12} T^{10} + \)\(12\!\cdots\!60\)\( p^{24} T^{12} - 512947080076 p^{36} T^{14} + p^{48} T^{16} )^{2} \) | |
| 71 | \( ( 1 - 608848232408 T^{2} + \)\(20\!\cdots\!88\)\( T^{4} - \)\(43\!\cdots\!16\)\( T^{6} + \)\(66\!\cdots\!70\)\( T^{8} - \)\(43\!\cdots\!16\)\( p^{12} T^{10} + \)\(20\!\cdots\!88\)\( p^{24} T^{12} - 608848232408 p^{36} T^{14} + p^{48} T^{16} )^{2} \) | |
| 73 | \( ( 1 - 462029561848 T^{2} + \)\(11\!\cdots\!48\)\( T^{4} - \)\(23\!\cdots\!36\)\( T^{6} + \)\(42\!\cdots\!70\)\( T^{8} - \)\(23\!\cdots\!36\)\( p^{12} T^{10} + \)\(11\!\cdots\!48\)\( p^{24} T^{12} - 462029561848 p^{36} T^{14} + p^{48} T^{16} )^{2} \) | |
| 79 | \( ( 1 + 353852 T + 404375732848 T^{2} + 127487920373701364 T^{3} + \)\(14\!\cdots\!70\)\( T^{4} + 127487920373701364 p^{6} T^{5} + 404375732848 p^{12} T^{6} + 353852 p^{18} T^{7} + p^{24} T^{8} )^{4} \) | |
| 83 | \( ( 1 - 120527747308 T^{2} + \)\(15\!\cdots\!68\)\( T^{4} + \)\(22\!\cdots\!04\)\( T^{6} + \)\(12\!\cdots\!70\)\( T^{8} + \)\(22\!\cdots\!04\)\( p^{12} T^{10} + \)\(15\!\cdots\!68\)\( p^{24} T^{12} - 120527747308 p^{36} T^{14} + p^{48} T^{16} )^{2} \) | |
| 89 | \( ( 1 + 132946313752 T^{2} + \)\(87\!\cdots\!48\)\( T^{4} + \)\(84\!\cdots\!64\)\( T^{6} + \)\(31\!\cdots\!70\)\( T^{8} + \)\(84\!\cdots\!64\)\( p^{12} T^{10} + \)\(87\!\cdots\!48\)\( p^{24} T^{12} + 132946313752 p^{36} T^{14} + p^{48} T^{16} )^{2} \) | |
| 97 | \( ( 1 - 4033397982136 T^{2} + \)\(83\!\cdots\!00\)\( T^{4} - \)\(11\!\cdots\!44\)\( T^{6} + \)\(11\!\cdots\!74\)\( T^{8} - \)\(11\!\cdots\!44\)\( p^{12} T^{10} + \)\(83\!\cdots\!00\)\( p^{24} T^{12} - 4033397982136 p^{36} T^{14} + p^{48} T^{16} )^{2} \) | |
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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
−2.00855376363097615715002955657, −2.00548775509511783327493241636, −1.97534063386358202929437411312, −1.95102613103113085067479575019, −1.92500758071220742688433178250, −1.88168268478412158549241245915, −1.71920539681120372683271813013, −1.64260548942010289260429727566, −1.49826332247757035357538202107, −1.37605560986223792489279003956, −1.35271152931887286452293609033, −1.19188239752363578298802666343, −1.03844125842734020404306535586, −0.900866902649644785581028642513, −0.842892693107338165808415481659, −0.800885301069242763600943247503, −0.75812283013459947379916580757, −0.74049255443242829364528883443, −0.71206894580728741394329521073, −0.53112761624686922856362839832, −0.49936037512200265300275375172, −0.48643960344778992303994324394, −0.31557057615446091937318084787, −0.15246632321768931633454443680, −0.13905517717868177836575238901, 0.13905517717868177836575238901, 0.15246632321768931633454443680, 0.31557057615446091937318084787, 0.48643960344778992303994324394, 0.49936037512200265300275375172, 0.53112761624686922856362839832, 0.71206894580728741394329521073, 0.74049255443242829364528883443, 0.75812283013459947379916580757, 0.800885301069242763600943247503, 0.842892693107338165808415481659, 0.900866902649644785581028642513, 1.03844125842734020404306535586, 1.19188239752363578298802666343, 1.35271152931887286452293609033, 1.37605560986223792489279003956, 1.49826332247757035357538202107, 1.64260548942010289260429727566, 1.71920539681120372683271813013, 1.88168268478412158549241245915, 1.92500758071220742688433178250, 1.95102613103113085067479575019, 1.97534063386358202929437411312, 2.00548775509511783327493241636, 2.00855376363097615715002955657
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.