Dirichlet series
| L(s) = 1 | − 5·2-s + 12·4-s − 6·5-s − 20·8-s − 11·9-s + 30·10-s − 10·13-s + 34·16-s − 10·17-s + 55·18-s − 72·20-s + 31·25-s + 50·26-s − 10·29-s − 65·32-s + 50·34-s − 132·36-s + 18·37-s + 120·40-s + 10·41-s + 66·45-s + 17·49-s − 155·50-s − 120·52-s + 38·53-s + 50·58-s − 10·61-s + ⋯ |
| L(s) = 1 | − 3.53·2-s + 6·4-s − 2.68·5-s − 7.07·8-s − 3.66·9-s + 9.48·10-s − 2.77·13-s + 17/2·16-s − 2.42·17-s + 12.9·18-s − 16.0·20-s + 31/5·25-s + 9.80·26-s − 1.85·29-s − 11.4·32-s + 8.57·34-s − 22·36-s + 2.95·37-s + 18.9·40-s + 1.56·41-s + 9.83·45-s + 17/7·49-s − 21.9·50-s − 16.6·52-s + 5.21·53-s + 6.56·58-s − 1.28·61-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{32} \cdot 11^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.39108\times 10^{-8}\) |
| Root analytic conductor: | \(0.592740\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{32} \cdot 11^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.003219240111\) |
| \(L(\frac12)\) | \(\approx\) | \(0.003219240111\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 + 5 T + 13 T^{2} + 25 T^{3} + 35 T^{4} + 15 p T^{5} - p T^{6} - 15 p^{2} T^{7} - 29 p^{2} T^{8} - 15 p^{3} T^{9} - p^{3} T^{10} + 15 p^{4} T^{11} + 35 p^{4} T^{12} + 25 p^{5} T^{13} + 13 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) |
| 11 | \( 1 - 35 T^{2} + 519 T^{4} - 5365 T^{6} + 5136 p T^{8} - 5365 p^{2} T^{10} + 519 p^{4} T^{12} - 35 p^{6} T^{14} + p^{8} T^{16} \) | |
| good | 3 | \( 1 + 11 T^{2} + 2 p^{3} T^{4} + 176 T^{6} + 2 p^{5} T^{8} + 491 p T^{10} + 659 p^{2} T^{12} + 23828 T^{14} + 78832 T^{16} + 23828 p^{2} T^{18} + 659 p^{6} T^{20} + 491 p^{7} T^{22} + 2 p^{13} T^{24} + 176 p^{10} T^{26} + 2 p^{15} T^{28} + 11 p^{14} T^{30} + p^{16} T^{32} \) |
| 5 | \( ( 1 + 3 T - 2 T^{2} - 14 T^{3} - 2 p T^{4} + 51 T^{5} + 153 T^{6} - 252 T^{7} - 1564 T^{8} - 252 p T^{9} + 153 p^{2} T^{10} + 51 p^{3} T^{11} - 2 p^{5} T^{12} - 14 p^{5} T^{13} - 2 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 7 | \( 1 - 17 T^{2} + 170 T^{4} - 160 p T^{6} + 3590 T^{8} - 5391 T^{10} + 51607 T^{12} - 1265760 T^{14} + 13240480 T^{16} - 1265760 p^{2} T^{18} + 51607 p^{4} T^{20} - 5391 p^{6} T^{22} + 3590 p^{8} T^{24} - 160 p^{11} T^{26} + 170 p^{12} T^{28} - 17 p^{14} T^{30} + p^{16} T^{32} \) | |
| 13 | \( ( 1 + 5 T + 42 T^{2} + 150 T^{3} + 620 T^{4} + 1455 T^{5} + 387 T^{6} - 3650 T^{7} - 65036 T^{8} - 3650 p T^{9} + 387 p^{2} T^{10} + 1455 p^{3} T^{11} + 620 p^{4} T^{12} + 150 p^{5} T^{13} + 42 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 17 | \( ( 1 + 5 T + 10 T^{2} + 486 T^{4} + 2555 T^{5} + 10495 T^{6} + 25600 T^{7} + 143056 T^{8} + 25600 p T^{9} + 10495 p^{2} T^{10} + 2555 p^{3} T^{11} + 486 p^{4} T^{12} + 10 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 19 | \( 1 + 30 T^{2} + 1233 T^{4} + 23510 T^{6} + 411393 T^{8} + 1849230 T^{10} - 90963349 T^{12} - 3587444450 T^{14} - 90905039420 T^{16} - 3587444450 p^{2} T^{18} - 90963349 p^{4} T^{20} + 1849230 p^{6} T^{22} + 411393 p^{8} T^{24} + 23510 p^{10} T^{26} + 1233 p^{12} T^{28} + 30 p^{14} T^{30} + p^{16} T^{32} \) | |
| 23 | \( ( 1 - 96 T^{2} + 4732 T^{4} - 162208 T^{6} + 4234310 T^{8} - 162208 p^{2} T^{10} + 4732 p^{4} T^{12} - 96 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 29 | \( ( 1 + 5 T + 78 T^{2} + 550 T^{3} + 3768 T^{4} + 26695 T^{5} + 166131 T^{6} + 945350 T^{7} + 5661780 T^{8} + 945350 p T^{9} + 166131 p^{2} T^{10} + 26695 p^{3} T^{11} + 3768 p^{4} T^{12} + 550 p^{5} T^{13} + 78 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 31 | \( 1 + 13 T^{2} + 46 T^{4} - 39168 T^{6} - 244734 T^{8} + 28347139 T^{10} + 776025319 T^{12} - 11750120144 T^{14} - 1558123543328 T^{16} - 11750120144 p^{2} T^{18} + 776025319 p^{4} T^{20} + 28347139 p^{6} T^{22} - 244734 p^{8} T^{24} - 39168 p^{10} T^{26} + 46 p^{12} T^{28} + 13 p^{14} T^{30} + p^{16} T^{32} \) | |
| 37 | \( ( 1 - 9 T + 58 T^{2} - 288 T^{3} + 2826 T^{4} - 8739 T^{5} + 5135 T^{6} + 158436 T^{7} + 832784 T^{8} + 158436 p T^{9} + 5135 p^{2} T^{10} - 8739 p^{3} T^{11} + 2826 p^{4} T^{12} - 288 p^{5} T^{13} + 58 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 41 | \( ( 1 - 5 T + 8 T^{2} - 540 T^{3} + 2928 T^{4} - 5515 T^{5} + 91991 T^{6} - 523800 T^{7} + 1936120 T^{8} - 523800 p T^{9} + 91991 p^{2} T^{10} - 5515 p^{3} T^{11} + 2928 p^{4} T^{12} - 540 p^{5} T^{13} + 8 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 43 | \( ( 1 + 201 T^{2} + 20307 T^{4} + 1361163 T^{6} + 67236960 T^{8} + 1361163 p^{2} T^{10} + 20307 p^{4} T^{12} + 201 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 47 | \( 1 + 59 T^{2} + 1174 T^{4} - 129116 T^{6} - 4850414 T^{8} + 212153137 T^{10} + 11841271291 T^{12} - 458606392528 T^{14} - 66560830358008 T^{16} - 458606392528 p^{2} T^{18} + 11841271291 p^{4} T^{20} + 212153137 p^{6} T^{22} - 4850414 p^{8} T^{24} - 129116 p^{10} T^{26} + 1174 p^{12} T^{28} + 59 p^{14} T^{30} + p^{16} T^{32} \) | |
| 53 | \( ( 1 - 19 T + 56 T^{2} + 1024 T^{3} - 7582 T^{4} + 24781 T^{5} - 179211 T^{6} - 1855738 T^{7} + 36671448 T^{8} - 1855738 p T^{9} - 179211 p^{2} T^{10} + 24781 p^{3} T^{11} - 7582 p^{4} T^{12} + 1024 p^{5} T^{13} + 56 p^{6} T^{14} - 19 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 59 | \( 1 + 350 T^{2} + 56233 T^{4} + 5299950 T^{6} + 306491273 T^{8} + 9575397350 T^{10} - 4883294509 T^{12} - 17205708063250 T^{14} - 1208178268282700 T^{16} - 17205708063250 p^{2} T^{18} - 4883294509 p^{4} T^{20} + 9575397350 p^{6} T^{22} + 306491273 p^{8} T^{24} + 5299950 p^{10} T^{26} + 56233 p^{12} T^{28} + 350 p^{14} T^{30} + p^{16} T^{32} \) | |
| 61 | \( ( 1 + 5 T - 4 T^{2} + 340 T^{3} + 6550 T^{4} + 21685 T^{5} + 106569 T^{6} + 2625650 T^{7} + 27744944 T^{8} + 2625650 p T^{9} + 106569 p^{2} T^{10} + 21685 p^{3} T^{11} + 6550 p^{4} T^{12} + 340 p^{5} T^{13} - 4 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 67 | \( ( 1 - 351 T^{2} + 61547 T^{4} - 6900053 T^{6} + 544631520 T^{8} - 6900053 p^{2} T^{10} + 61547 p^{4} T^{12} - 351 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 71 | \( 1 + 285 T^{2} + 39458 T^{4} + 3119420 T^{6} + 118131358 T^{8} + 404779135 T^{10} + 205412628291 T^{12} + 73614429095400 T^{14} + 7731123817186280 T^{16} + 73614429095400 p^{2} T^{18} + 205412628291 p^{4} T^{20} + 404779135 p^{6} T^{22} + 118131358 p^{8} T^{24} + 3119420 p^{10} T^{26} + 39458 p^{12} T^{28} + 285 p^{14} T^{30} + p^{16} T^{32} \) | |
| 73 | \( ( 1 + 15 T + 182 T^{2} + 960 T^{3} + 8490 T^{4} + 51585 T^{5} + 875047 T^{6} + 4632960 T^{7} + 56899184 T^{8} + 4632960 p T^{9} + 875047 p^{2} T^{10} + 51585 p^{3} T^{11} + 8490 p^{4} T^{12} + 960 p^{5} T^{13} + 182 p^{6} T^{14} + 15 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 79 | \( 1 - 175 T^{2} + 25018 T^{4} - 2721680 T^{6} + 276506938 T^{8} - 24540773525 T^{10} + 2169836003371 T^{12} - 177413374417900 T^{14} + 14750049423098000 T^{16} - 177413374417900 p^{2} T^{18} + 2169836003371 p^{4} T^{20} - 24540773525 p^{6} T^{22} + 276506938 p^{8} T^{24} - 2721680 p^{10} T^{26} + 25018 p^{12} T^{28} - 175 p^{14} T^{30} + p^{16} T^{32} \) | |
| 83 | \( 1 - 432 T^{2} + 87375 T^{4} - 11521130 T^{6} + 1183958625 T^{8} - 112754321886 T^{10} + 11335103772317 T^{12} - 1139715862795000 T^{14} + 102156150284526820 T^{16} - 1139715862795000 p^{2} T^{18} + 11335103772317 p^{4} T^{20} - 112754321886 p^{6} T^{22} + 1183958625 p^{8} T^{24} - 11521130 p^{10} T^{26} + 87375 p^{12} T^{28} - 432 p^{14} T^{30} + p^{16} T^{32} \) | |
| 89 | \( ( 1 + 9 T + 337 T^{2} + 2217 T^{3} + 44260 T^{4} + 2217 p T^{5} + 337 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 97 | \( ( 1 + 17 T + 87 T^{2} + 1195 T^{3} + 20696 T^{4} + 1195 p T^{5} + 87 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
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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
−5.48796192545704803665547124242, −5.42921536950679246297502834054, −5.38726880851750830540119245571, −5.02909577387759671992870111748, −4.98840083919941029063903722522, −4.77211385045028668663464353930, −4.65681933478518202795766550077, −4.60794759224348415695938603222, −4.54091099639989253225350124466, −4.21844425640318850237605637302, −4.13958081293968062323330834959, −4.12555725224630642994296772726, −3.88494080891157511628563271071, −3.77428715166864530891938463609, −3.76763754896625893689228736800, −3.29789081753004457615719220340, −3.20137025099284401050165051776, −3.16150572257945797452774858010, −2.76082777466018957972996401558, −2.61795708095079875935400015112, −2.60254900193392084902177184120, −2.58412155340676958706680891206, −2.43970731718250358503347518610, −2.26086986162037297193259723631, −1.30057736695860585162283973664, 1.30057736695860585162283973664, 2.26086986162037297193259723631, 2.43970731718250358503347518610, 2.58412155340676958706680891206, 2.60254900193392084902177184120, 2.61795708095079875935400015112, 2.76082777466018957972996401558, 3.16150572257945797452774858010, 3.20137025099284401050165051776, 3.29789081753004457615719220340, 3.76763754896625893689228736800, 3.77428715166864530891938463609, 3.88494080891157511628563271071, 4.12555725224630642994296772726, 4.13958081293968062323330834959, 4.21844425640318850237605637302, 4.54091099639989253225350124466, 4.60794759224348415695938603222, 4.65681933478518202795766550077, 4.77211385045028668663464353930, 4.98840083919941029063903722522, 5.02909577387759671992870111748, 5.38726880851750830540119245571, 5.42921536950679246297502834054, 5.48796192545704803665547124242
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.