Dirichlet series
| L(s) = 1 | − 6·5-s − 2·7-s + 332·9-s − 310·19-s − 3.48e3·25-s + 734·31-s + 12·35-s − 210·41-s − 1.99e3·45-s − 1.99e3·47-s − 1.19e4·49-s + 5.61e3·59-s − 664·63-s − 5.42e3·67-s − 1.73e3·71-s + 5.50e4·81-s + 1.86e3·95-s + 2.94e3·97-s + 2.15e3·101-s − 3.90e3·103-s + 1.12e3·107-s − 1.81e4·109-s + 1.92e4·113-s + 8.44e4·121-s + 1.94e4·125-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 0.239·5-s − 0.0408·7-s + 4.09·9-s − 0.858·19-s − 5.57·25-s + 0.763·31-s + 0.00979·35-s − 0.124·41-s − 0.983·45-s − 0.901·47-s − 4.96·49-s + 1.61·59-s − 0.167·63-s − 1.20·67-s − 0.343·71-s + 8.39·81-s + 0.206·95-s + 0.312·97-s + 0.211·101-s − 0.367·103-s + 0.0979·107-s − 1.52·109-s + 1.50·113-s + 5.76·121-s + 1.24·125-s + 6.20e−5·127-s + 5.82e−5·131-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 31^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 31^{10}\right)^{s/2} \, \Gamma_{\C}(s+2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(2^{20} \cdot 31^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.19718\times 10^{11}\) |
| Root analytic conductor: | \(3.58020\) |
| Motivic weight: | \(4\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 2^{20} \cdot 31^{10} ,\ ( \ : [2]^{10} ),\ 1 )\) |
Particular Values
| \(L(\frac{5}{2})\) | \(\approx\) | \(8.675122059\) |
| \(L(\frac12)\) | \(\approx\) | \(8.675122059\) |
| \(L(3)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 31 | \( 1 - 734 T + 74027 p T^{2} - 1058616 p^{2} T^{3} + 104554486 p^{3} T^{4} - 47359404 p^{5} T^{5} + 104554486 p^{7} T^{6} - 1058616 p^{10} T^{7} + 74027 p^{13} T^{8} - 734 p^{16} T^{9} + p^{20} T^{10} \) | |
| good | 3 | \( 1 - 332 T^{2} + 55169 T^{4} - 1873648 p T^{6} + 15246746 p^{3} T^{8} - 120731864 p^{5} T^{10} + 15246746 p^{11} T^{12} - 1873648 p^{17} T^{14} + 55169 p^{24} T^{16} - 332 p^{32} T^{18} + p^{40} T^{20} \) |
| 5 | \( ( 1 + 3 T + 1756 T^{2} + 1209 p T^{3} + 1609679 T^{4} + 5692488 T^{5} + 1609679 p^{4} T^{6} + 1209 p^{9} T^{7} + 1756 p^{12} T^{8} + 3 p^{16} T^{9} + p^{20} T^{10} )^{2} \) | |
| 7 | \( ( 1 + T + 5956 T^{2} - 21099 T^{3} + 20441753 T^{4} - 39615256 T^{5} + 20441753 p^{4} T^{6} - 21099 p^{8} T^{7} + 5956 p^{12} T^{8} + p^{16} T^{9} + p^{20} T^{10} )^{2} \) | |
| 11 | \( 1 - 84416 T^{2} + 3032365933 T^{4} - 60492289539072 T^{6} + 6470173125812354 p^{2} T^{8} - 656507644698325888 p^{4} T^{10} + 6470173125812354 p^{10} T^{12} - 60492289539072 p^{16} T^{14} + 3032365933 p^{24} T^{16} - 84416 p^{32} T^{18} + p^{40} T^{20} \) | |
| 13 | \( 1 - 173268 T^{2} + 14503834545 T^{4} - 786779593736400 T^{6} + 31475634602869952142 T^{8} - \)\(99\!\cdots\!08\)\( T^{10} + 31475634602869952142 p^{8} T^{12} - 786779593736400 p^{16} T^{14} + 14503834545 p^{24} T^{16} - 173268 p^{32} T^{18} + p^{40} T^{20} \) | |
| 17 | \( 1 - 562382 T^{2} + 145316748601 T^{4} - 23296071343768944 T^{6} + \)\(26\!\cdots\!70\)\( T^{8} - \)\(24\!\cdots\!20\)\( T^{10} + \)\(26\!\cdots\!70\)\( p^{8} T^{12} - 23296071343768944 p^{16} T^{14} + 145316748601 p^{24} T^{16} - 562382 p^{32} T^{18} + p^{40} T^{20} \) | |
| 19 | \( ( 1 + 155 T + 10802 p T^{2} + 87003395 T^{3} + 33357697489 T^{4} + 15429976759380 T^{5} + 33357697489 p^{4} T^{6} + 87003395 p^{8} T^{7} + 10802 p^{13} T^{8} + 155 p^{16} T^{9} + p^{20} T^{10} )^{2} \) | |
| 23 | \( 1 - 1393710 T^{2} + 1077623343417 T^{4} - 580886777256586896 T^{6} + \)\(23\!\cdots\!62\)\( T^{8} - \)\(74\!\cdots\!40\)\( T^{10} + \)\(23\!\cdots\!62\)\( p^{8} T^{12} - 580886777256586896 p^{16} T^{14} + 1077623343417 p^{24} T^{16} - 1393710 p^{32} T^{18} + p^{40} T^{20} \) | |
| 29 | \( 1 - 172596 T^{2} + 1498323855921 T^{4} - 644825281382866896 T^{6} + \)\(97\!\cdots\!38\)\( T^{8} - \)\(55\!\cdots\!04\)\( T^{10} + \)\(97\!\cdots\!38\)\( p^{8} T^{12} - 644825281382866896 p^{16} T^{14} + 1498323855921 p^{24} T^{16} - 172596 p^{32} T^{18} + p^{40} T^{20} \) | |
| 37 | \( 1 - 6673840 T^{2} + 21421105937517 T^{4} - 51973611938665455168 T^{6} + \)\(12\!\cdots\!30\)\( T^{8} - \)\(27\!\cdots\!48\)\( T^{10} + \)\(12\!\cdots\!30\)\( p^{8} T^{12} - 51973611938665455168 p^{16} T^{14} + 21421105937517 p^{24} T^{16} - 6673840 p^{32} T^{18} + p^{40} T^{20} \) | |
| 41 | \( ( 1 + 105 T + 9012530 T^{2} - 1330195545 T^{3} + 37910018689055 T^{4} - 9215870425913748 T^{5} + 37910018689055 p^{4} T^{6} - 1330195545 p^{8} T^{7} + 9012530 p^{12} T^{8} + 105 p^{16} T^{9} + p^{20} T^{10} )^{2} \) | |
| 43 | \( 1 - 8827772 T^{2} + 51198724542721 T^{4} - \)\(22\!\cdots\!04\)\( T^{6} + \)\(93\!\cdots\!50\)\( T^{8} - \)\(33\!\cdots\!60\)\( T^{10} + \)\(93\!\cdots\!50\)\( p^{8} T^{12} - \)\(22\!\cdots\!04\)\( p^{16} T^{14} + 51198724542721 p^{24} T^{16} - 8827772 p^{32} T^{18} + p^{40} T^{20} \) | |
| 47 | \( ( 1 + 996 T + 19069589 T^{2} + 21966753744 T^{3} + 160612850425322 T^{4} + 166831416876206616 T^{5} + 160612850425322 p^{4} T^{6} + 21966753744 p^{8} T^{7} + 19069589 p^{12} T^{8} + 996 p^{16} T^{9} + p^{20} T^{10} )^{2} \) | |
| 53 | \( 1 - 46388448 T^{2} + 1102207214878989 T^{4} - \)\(17\!\cdots\!56\)\( T^{6} + \)\(20\!\cdots\!82\)\( T^{8} - \)\(18\!\cdots\!68\)\( T^{10} + \)\(20\!\cdots\!82\)\( p^{8} T^{12} - \)\(17\!\cdots\!56\)\( p^{16} T^{14} + 1102207214878989 p^{24} T^{16} - 46388448 p^{32} T^{18} + p^{40} T^{20} \) | |
| 59 | \( ( 1 - 2805 T + 32571222 T^{2} - 74954161089 T^{3} + 578048385399557 T^{4} - 949272226602352620 T^{5} + 578048385399557 p^{4} T^{6} - 74954161089 p^{8} T^{7} + 32571222 p^{12} T^{8} - 2805 p^{16} T^{9} + p^{20} T^{10} )^{2} \) | |
| 61 | \( 1 - 63441332 T^{2} + 2146219603773937 T^{4} - \)\(83\!\cdots\!72\)\( p T^{6} + \)\(95\!\cdots\!86\)\( T^{8} - \)\(14\!\cdots\!52\)\( T^{10} + \)\(95\!\cdots\!86\)\( p^{8} T^{12} - \)\(83\!\cdots\!72\)\( p^{17} T^{14} + 2146219603773937 p^{24} T^{16} - 63441332 p^{32} T^{18} + p^{40} T^{20} \) | |
| 67 | \( ( 1 + 2710 T + 44680697 T^{2} + 46030758192 T^{3} + 1293620881955350 T^{4} + 1520957394592633332 T^{5} + 1293620881955350 p^{4} T^{6} + 46030758192 p^{8} T^{7} + 44680697 p^{12} T^{8} + 2710 p^{16} T^{9} + p^{20} T^{10} )^{2} \) | |
| 71 | \( ( 1 + 867 T + 74387622 T^{2} - 93427444953 T^{3} + 2408453745254357 T^{4} - 5672151635204928588 T^{5} + 2408453745254357 p^{4} T^{6} - 93427444953 p^{8} T^{7} + 74387622 p^{12} T^{8} + 867 p^{16} T^{9} + p^{20} T^{10} )^{2} \) | |
| 73 | \( 1 - 175152578 T^{2} + 15304038194700589 T^{4} - \)\(88\!\cdots\!76\)\( T^{6} + \)\(37\!\cdots\!22\)\( T^{8} - \)\(12\!\cdots\!28\)\( T^{10} + \)\(37\!\cdots\!22\)\( p^{8} T^{12} - \)\(88\!\cdots\!76\)\( p^{16} T^{14} + 15304038194700589 p^{24} T^{16} - 175152578 p^{32} T^{18} + p^{40} T^{20} \) | |
| 79 | \( 1 - 205133046 T^{2} + 23575039387323993 T^{4} - \)\(18\!\cdots\!52\)\( T^{6} + \)\(10\!\cdots\!94\)\( T^{8} - \)\(46\!\cdots\!68\)\( T^{10} + \)\(10\!\cdots\!94\)\( p^{8} T^{12} - \)\(18\!\cdots\!52\)\( p^{16} T^{14} + 23575039387323993 p^{24} T^{16} - 205133046 p^{32} T^{18} + p^{40} T^{20} \) | |
| 83 | \( 1 - 191562000 T^{2} + 20412146555701197 T^{4} - \)\(15\!\cdots\!28\)\( T^{6} + \)\(99\!\cdots\!30\)\( T^{8} - \)\(52\!\cdots\!08\)\( T^{10} + \)\(99\!\cdots\!30\)\( p^{8} T^{12} - \)\(15\!\cdots\!28\)\( p^{16} T^{14} + 20412146555701197 p^{24} T^{16} - 191562000 p^{32} T^{18} + p^{40} T^{20} \) | |
| 89 | \( 1 - 361321382 T^{2} + 67523456731169497 T^{4} - \)\(84\!\cdots\!72\)\( T^{6} + \)\(78\!\cdots\!46\)\( T^{8} - \)\(55\!\cdots\!72\)\( T^{10} + \)\(78\!\cdots\!46\)\( p^{8} T^{12} - \)\(84\!\cdots\!72\)\( p^{16} T^{14} + 67523456731169497 p^{24} T^{16} - 361321382 p^{32} T^{18} + p^{40} T^{20} \) | |
| 97 | \( ( 1 - 1471 T + 226470674 T^{2} + 834470260179 T^{3} + 22628652492129979 T^{4} + \)\(15\!\cdots\!52\)\( T^{5} + 22628652492129979 p^{4} T^{6} + 834470260179 p^{8} T^{7} + 226470674 p^{12} T^{8} - 1471 p^{16} T^{9} + p^{20} T^{10} )^{2} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.46961884171268848193165269879, −4.23598389579232752053403421203, −4.22363333909705277466061567965, −4.18275611680935565356227564771, −4.13842788472156738342984316590, −4.03950673462484467389761341631, −3.62293359438886084606599589571, −3.45489419281473231269267669073, −3.37770274580925805380225089120, −3.26593284226814587538555079985, −3.22309055484746769502701560409, −2.94742046102025515406929858895, −2.52172281367231965411763856501, −2.27860280041002827827254035919, −2.23152759880245786631432386610, −1.87258333600320374393363969336, −1.85116505600354256105250398079, −1.77153200144351860556812464767, −1.54157197554776294718236694487, −1.41004190545981392404428206207, −1.35220660067368787571104538471, −0.70508037484400056570609170393, −0.56317554929320268358147586530, −0.42984351090951568179311212496, −0.23778300624461973480459828056, 0.23778300624461973480459828056, 0.42984351090951568179311212496, 0.56317554929320268358147586530, 0.70508037484400056570609170393, 1.35220660067368787571104538471, 1.41004190545981392404428206207, 1.54157197554776294718236694487, 1.77153200144351860556812464767, 1.85116505600354256105250398079, 1.87258333600320374393363969336, 2.23152759880245786631432386610, 2.27860280041002827827254035919, 2.52172281367231965411763856501, 2.94742046102025515406929858895, 3.22309055484746769502701560409, 3.26593284226814587538555079985, 3.37770274580925805380225089120, 3.45489419281473231269267669073, 3.62293359438886084606599589571, 4.03950673462484467389761341631, 4.13842788472156738342984316590, 4.18275611680935565356227564771, 4.22363333909705277466061567965, 4.23598389579232752053403421203, 4.46961884171268848193165269879
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.