Dirichlet series
| L(s) = 1 | + 25·3-s + 503·4-s − 1.96e3·7-s − 5.29e3·9-s + 1.25e4·12-s + 1.69e4·13-s + 3.82e4·16-s − 1.43e5·19-s − 4.90e4·21-s − 5.05e4·27-s − 9.85e5·28-s − 3.01e6·31-s − 2.66e6·36-s − 3.01e6·37-s + 4.23e5·39-s − 1.17e7·43-s + 9.55e5·48-s − 2.24e7·49-s + 8.51e6·52-s − 3.59e6·57-s + 1.52e6·61-s + 1.03e7·63-s − 2.20e7·64-s + 1.62e7·67-s + 5.20e7·73-s − 7.23e7·76-s + 8.54e6·79-s + ⋯ |
| L(s) = 1 | + 0.308·3-s + 1.96·4-s − 0.816·7-s − 0.806·9-s + 0.606·12-s + 0.592·13-s + 0.582·16-s − 1.10·19-s − 0.251·21-s − 0.0950·27-s − 1.60·28-s − 3.26·31-s − 1.58·36-s − 1.60·37-s + 0.182·39-s − 3.43·43-s + 0.179·48-s − 3.89·49-s + 1.16·52-s − 0.340·57-s + 0.109·61-s + 0.658·63-s − 1.31·64-s + 0.807·67-s + 1.83·73-s − 2.17·76-s + 0.219·79-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 5^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 5^{20}\right)^{s/2} \, \Gamma_{\C}(s+4)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(3^{10} \cdot 5^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.08916\times 10^{14}\) |
| Root analytic conductor: | \(5.52751\) |
| Motivic weight: | \(8\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 3^{10} \cdot 5^{20} ,\ ( \ : [4]^{10} ),\ 1 )\) |
Particular Values
| \(L(\frac{9}{2})\) | \(\approx\) | \(1.804183252\) |
| \(L(\frac12)\) | \(\approx\) | \(1.804183252\) |
| \(L(5)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 - 25 T + 1972 p T^{2} - 35 p^{8} T^{3} + 19187 p^{6} T^{4} - 1761160 p^{7} T^{5} + 19187 p^{14} T^{6} - 35 p^{24} T^{7} + 1972 p^{25} T^{8} - 25 p^{32} T^{9} + p^{40} T^{10} \) |
| 5 | \( 1 \) | |
| good | 2 | \( 1 - 503 T^{2} + 107401 p T^{4} - 2086639 p^{5} T^{6} + 138411569 p^{7} T^{8} - 152456901 p^{15} T^{10} + 138411569 p^{23} T^{12} - 2086639 p^{37} T^{14} + 107401 p^{49} T^{16} - 503 p^{64} T^{18} + p^{80} T^{20} \) |
| 7 | \( ( 1 + 20 p^{2} T + 1807688 p T^{2} + 605153480 p^{2} T^{3} + 178140304501 p^{3} T^{4} + 115043780623080 p^{4} T^{5} + 178140304501 p^{11} T^{6} + 605153480 p^{18} T^{7} + 1807688 p^{25} T^{8} + 20 p^{34} T^{9} + p^{40} T^{10} )^{2} \) | |
| 11 | \( 1 - 192340745 T^{2} + 148881111120899000 T^{4} - \)\(33\!\cdots\!75\)\( T^{6} + \)\(10\!\cdots\!95\)\( T^{8} - \)\(22\!\cdots\!52\)\( T^{10} + \)\(10\!\cdots\!95\)\( p^{16} T^{12} - \)\(33\!\cdots\!75\)\( p^{32} T^{14} + 148881111120899000 p^{48} T^{16} - 192340745 p^{64} T^{18} + p^{80} T^{20} \) | |
| 13 | \( ( 1 - 8460 T + 2061295726 T^{2} - 19630735647410 T^{3} + 2703420250858988833 T^{4} - \)\(21\!\cdots\!00\)\( T^{5} + 2703420250858988833 p^{8} T^{6} - 19630735647410 p^{16} T^{7} + 2061295726 p^{24} T^{8} - 8460 p^{32} T^{9} + p^{40} T^{10} )^{2} \) | |
| 17 | \( 1 - 24622202753 T^{2} + \)\(30\!\cdots\!72\)\( T^{4} - \)\(26\!\cdots\!23\)\( T^{6} + \)\(21\!\cdots\!07\)\( T^{8} - \)\(15\!\cdots\!08\)\( T^{10} + \)\(21\!\cdots\!07\)\( p^{16} T^{12} - \)\(26\!\cdots\!23\)\( p^{32} T^{14} + \)\(30\!\cdots\!72\)\( p^{48} T^{16} - 24622202753 p^{64} T^{18} + p^{80} T^{20} \) | |
| 19 | \( ( 1 + 71967 T + 50409342607 T^{2} + 4310541913802 p^{2} T^{3} + \)\(11\!\cdots\!57\)\( T^{4} + \)\(16\!\cdots\!17\)\( T^{5} + \)\(11\!\cdots\!57\)\( p^{8} T^{6} + 4310541913802 p^{18} T^{7} + 50409342607 p^{24} T^{8} + 71967 p^{32} T^{9} + p^{40} T^{10} )^{2} \) | |
| 23 | \( 1 - 502916724718 T^{2} + \)\(12\!\cdots\!77\)\( T^{4} - \)\(20\!\cdots\!88\)\( T^{6} + \)\(24\!\cdots\!82\)\( T^{8} - \)\(21\!\cdots\!08\)\( T^{10} + \)\(24\!\cdots\!82\)\( p^{16} T^{12} - \)\(20\!\cdots\!88\)\( p^{32} T^{14} + \)\(12\!\cdots\!77\)\( p^{48} T^{16} - 502916724718 p^{64} T^{18} + p^{80} T^{20} \) | |
| 29 | \( 1 - 2741150439950 T^{2} + \)\(40\!\cdots\!65\)\( T^{4} - \)\(40\!\cdots\!40\)\( T^{6} + \)\(30\!\cdots\!50\)\( T^{8} - \)\(17\!\cdots\!52\)\( T^{10} + \)\(30\!\cdots\!50\)\( p^{16} T^{12} - \)\(40\!\cdots\!40\)\( p^{32} T^{14} + \)\(40\!\cdots\!65\)\( p^{48} T^{16} - 2741150439950 p^{64} T^{18} + p^{80} T^{20} \) | |
| 31 | \( ( 1 + 1507030 T + 4095821792900 T^{2} + 4409014857553613950 T^{3} + \)\(67\!\cdots\!95\)\( T^{4} + \)\(53\!\cdots\!48\)\( T^{5} + \)\(67\!\cdots\!95\)\( p^{8} T^{6} + 4409014857553613950 p^{16} T^{7} + 4095821792900 p^{24} T^{8} + 1507030 p^{32} T^{9} + p^{40} T^{10} )^{2} \) | |
| 37 | \( ( 1 + 1508380 T + 10316940851001 T^{2} + 10554349546069507440 T^{3} + \)\(44\!\cdots\!58\)\( T^{4} + \)\(37\!\cdots\!20\)\( T^{5} + \)\(44\!\cdots\!58\)\( p^{8} T^{6} + 10554349546069507440 p^{16} T^{7} + 10316940851001 p^{24} T^{8} + 1508380 p^{32} T^{9} + p^{40} T^{10} )^{2} \) | |
| 41 | \( 1 - 41649195595145 T^{2} + \)\(90\!\cdots\!00\)\( T^{4} - \)\(13\!\cdots\!75\)\( T^{6} + \)\(15\!\cdots\!95\)\( T^{8} - \)\(14\!\cdots\!52\)\( T^{10} + \)\(15\!\cdots\!95\)\( p^{16} T^{12} - \)\(13\!\cdots\!75\)\( p^{32} T^{14} + \)\(90\!\cdots\!00\)\( p^{48} T^{16} - 41649195595145 p^{64} T^{18} + p^{80} T^{20} \) | |
| 43 | \( ( 1 + 5873670 T + 55623037219636 T^{2} + \)\(23\!\cdots\!30\)\( T^{3} + \)\(12\!\cdots\!03\)\( T^{4} + \)\(38\!\cdots\!20\)\( T^{5} + \)\(12\!\cdots\!03\)\( p^{8} T^{6} + \)\(23\!\cdots\!30\)\( p^{16} T^{7} + 55623037219636 p^{24} T^{8} + 5873670 p^{32} T^{9} + p^{40} T^{10} )^{2} \) | |
| 47 | \( 1 - 94478339945798 T^{2} + \)\(45\!\cdots\!77\)\( T^{4} - \)\(14\!\cdots\!88\)\( T^{6} + \)\(35\!\cdots\!42\)\( T^{8} - \)\(82\!\cdots\!68\)\( T^{10} + \)\(35\!\cdots\!42\)\( p^{16} T^{12} - \)\(14\!\cdots\!88\)\( p^{32} T^{14} + \)\(45\!\cdots\!77\)\( p^{48} T^{16} - 94478339945798 p^{64} T^{18} + p^{80} T^{20} \) | |
| 53 | \( 1 - 314820558259598 T^{2} + \)\(51\!\cdots\!77\)\( T^{4} - \)\(10\!\cdots\!96\)\( p T^{6} + \)\(44\!\cdots\!42\)\( T^{8} - \)\(30\!\cdots\!68\)\( T^{10} + \)\(44\!\cdots\!42\)\( p^{16} T^{12} - \)\(10\!\cdots\!96\)\( p^{33} T^{14} + \)\(51\!\cdots\!77\)\( p^{48} T^{16} - 314820558259598 p^{64} T^{18} + p^{80} T^{20} \) | |
| 59 | \( 1 - 543628475970170 T^{2} + \)\(20\!\cdots\!25\)\( T^{4} - \)\(54\!\cdots\!00\)\( T^{6} + \)\(11\!\cdots\!70\)\( T^{8} - \)\(18\!\cdots\!52\)\( T^{10} + \)\(11\!\cdots\!70\)\( p^{16} T^{12} - \)\(54\!\cdots\!00\)\( p^{32} T^{14} + \)\(20\!\cdots\!25\)\( p^{48} T^{16} - 543628475970170 p^{64} T^{18} + p^{80} T^{20} \) | |
| 61 | \( ( 1 - 760110 T + 696303715647670 T^{2} - \)\(12\!\cdots\!20\)\( T^{3} + \)\(22\!\cdots\!85\)\( T^{4} - \)\(38\!\cdots\!52\)\( T^{5} + \)\(22\!\cdots\!85\)\( p^{8} T^{6} - \)\(12\!\cdots\!20\)\( p^{16} T^{7} + 696303715647670 p^{24} T^{8} - 760110 p^{32} T^{9} + p^{40} T^{10} )^{2} \) | |
| 67 | \( ( 1 - 8134645 T + 1108061463006911 T^{2} - \)\(13\!\cdots\!90\)\( T^{3} + \)\(75\!\cdots\!73\)\( T^{4} - \)\(67\!\cdots\!15\)\( T^{5} + \)\(75\!\cdots\!73\)\( p^{8} T^{6} - \)\(13\!\cdots\!90\)\( p^{16} T^{7} + 1108061463006911 p^{24} T^{8} - 8134645 p^{32} T^{9} + p^{40} T^{10} )^{2} \) | |
| 71 | \( 1 - 2732196231020750 T^{2} + \)\(39\!\cdots\!65\)\( T^{4} - \)\(41\!\cdots\!40\)\( T^{6} + \)\(36\!\cdots\!50\)\( T^{8} - \)\(25\!\cdots\!52\)\( T^{10} + \)\(36\!\cdots\!50\)\( p^{16} T^{12} - \)\(41\!\cdots\!40\)\( p^{32} T^{14} + \)\(39\!\cdots\!65\)\( p^{48} T^{16} - 2732196231020750 p^{64} T^{18} + p^{80} T^{20} \) | |
| 73 | \( ( 1 - 26045085 T + 18917014698616 T^{2} + \)\(27\!\cdots\!05\)\( T^{3} + \)\(59\!\cdots\!83\)\( T^{4} - \)\(32\!\cdots\!20\)\( T^{5} + \)\(59\!\cdots\!83\)\( p^{8} T^{6} + \)\(27\!\cdots\!05\)\( p^{16} T^{7} + 18917014698616 p^{24} T^{8} - 26045085 p^{32} T^{9} + p^{40} T^{10} )^{2} \) | |
| 79 | \( ( 1 - 4274948 T + 4956420190243457 T^{2} - \)\(17\!\cdots\!08\)\( T^{3} + \)\(12\!\cdots\!02\)\( T^{4} - \)\(32\!\cdots\!08\)\( T^{5} + \)\(12\!\cdots\!02\)\( p^{8} T^{6} - \)\(17\!\cdots\!08\)\( p^{16} T^{7} + 4956420190243457 p^{24} T^{8} - 4274948 p^{32} T^{9} + p^{40} T^{10} )^{2} \) | |
| 83 | \( 1 - 11035625920352113 T^{2} + \)\(66\!\cdots\!72\)\( T^{4} - \)\(27\!\cdots\!83\)\( T^{6} + \)\(87\!\cdots\!07\)\( T^{8} - \)\(21\!\cdots\!68\)\( T^{10} + \)\(87\!\cdots\!07\)\( p^{16} T^{12} - \)\(27\!\cdots\!83\)\( p^{32} T^{14} + \)\(66\!\cdots\!72\)\( p^{48} T^{16} - 11035625920352113 p^{64} T^{18} + p^{80} T^{20} \) | |
| 89 | \( 1 - 21967942241494945 T^{2} + \)\(22\!\cdots\!00\)\( T^{4} - \)\(13\!\cdots\!75\)\( T^{6} + \)\(64\!\cdots\!95\)\( T^{8} - \)\(26\!\cdots\!52\)\( T^{10} + \)\(64\!\cdots\!95\)\( p^{16} T^{12} - \)\(13\!\cdots\!75\)\( p^{32} T^{14} + \)\(22\!\cdots\!00\)\( p^{48} T^{16} - 21967942241494945 p^{64} T^{18} + p^{80} T^{20} \) | |
| 97 | \( ( 1 - 201083900 T + 45058329689335806 T^{2} - \)\(58\!\cdots\!90\)\( T^{3} + \)\(74\!\cdots\!93\)\( T^{4} - \)\(66\!\cdots\!80\)\( T^{5} + \)\(74\!\cdots\!93\)\( p^{8} T^{6} - \)\(58\!\cdots\!90\)\( p^{16} T^{7} + 45058329689335806 p^{24} T^{8} - 201083900 p^{32} T^{9} + p^{40} T^{10} )^{2} \) | |
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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
−4.30235591554057914210862043349, −3.79454320040743435147952472864, −3.62139701994354469031339335486, −3.49144950422100285509492603347, −3.44544693627890888442565112772, −3.43512913205131528841713850350, −3.13644420483490294925533880737, −3.08651729851563801221615894075, −3.03434458691381823941199012528, −3.01120167836475632409913428985, −2.40154493689585325778227554911, −2.22676871069643246577506390675, −2.09846209206546511926515277591, −1.94296137678566180303261440217, −1.91892513911487202599344174671, −1.84755634522310508657643014811, −1.84206930308751725278704436031, −1.69748173813427837275796391554, −1.25243919009494480060137379185, −0.902568153977889926325433794099, −0.72698544528957447495255407896, −0.68577879199553175308852857589, −0.49689369784829477081516908870, −0.17313488402755529894851595706, −0.11336215254886997345261706357, 0.11336215254886997345261706357, 0.17313488402755529894851595706, 0.49689369784829477081516908870, 0.68577879199553175308852857589, 0.72698544528957447495255407896, 0.902568153977889926325433794099, 1.25243919009494480060137379185, 1.69748173813427837275796391554, 1.84206930308751725278704436031, 1.84755634522310508657643014811, 1.91892513911487202599344174671, 1.94296137678566180303261440217, 2.09846209206546511926515277591, 2.22676871069643246577506390675, 2.40154493689585325778227554911, 3.01120167836475632409913428985, 3.03434458691381823941199012528, 3.08651729851563801221615894075, 3.13644420483490294925533880737, 3.43512913205131528841713850350, 3.44544693627890888442565112772, 3.49144950422100285509492603347, 3.62139701994354469031339335486, 3.79454320040743435147952472864, 4.30235591554057914210862043349
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.