Dirichlet series
| L(s) = 1 | − 10·4-s − 18·9-s − 22·11-s + 39·16-s − 28·19-s − 20·29-s − 32·31-s + 180·36-s − 52·41-s + 220·44-s − 41·49-s − 76·59-s + 8·61-s − 64·64-s − 72·71-s + 280·76-s + 16·79-s + 141·81-s − 94·89-s + 396·99-s + 30·101-s − 18·109-s + 200·116-s + 167·121-s + 320·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 5·4-s − 6·9-s − 6.63·11-s + 39/4·16-s − 6.42·19-s − 3.71·29-s − 5.74·31-s + 30·36-s − 8.12·41-s + 33.1·44-s − 5.85·49-s − 9.89·59-s + 1.02·61-s − 8·64-s − 8.54·71-s + 32.1·76-s + 1.80·79-s + 47/3·81-s − 9.96·89-s + 39.7·99-s + 2.98·101-s − 1.72·109-s + 18.5·116-s + 15.1·121-s + 28.7·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{36} \cdot 13^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{36} \cdot 13^{36}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(36\) |
| Conductor: | \(5^{36} \cdot 13^{36}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.20530\times 10^{27}\) |
| Root analytic conductor: | \(5.80833\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(18\) |
| Selberg data: | \((36,\ 5^{36} \cdot 13^{36} ,\ ( \ : [1/2]^{18} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 \) | |
| good | 2 | \( 1 + 5 p T^{2} + 61 T^{4} + 71 p^{2} T^{6} + 539 p T^{8} + 875 p^{2} T^{10} + 2497 p^{2} T^{12} + 25443 T^{14} + 1831 p^{5} T^{16} + 122763 T^{18} + 1831 p^{7} T^{20} + 25443 p^{4} T^{22} + 2497 p^{8} T^{24} + 875 p^{10} T^{26} + 539 p^{11} T^{28} + 71 p^{14} T^{30} + 61 p^{14} T^{32} + 5 p^{17} T^{34} + p^{18} T^{36} \) |
| 3 | \( 1 + 2 p^{2} T^{2} + 61 p T^{4} + 1357 T^{6} + 8149 T^{8} + 41485 T^{10} + 183952 T^{12} + 722131 T^{14} + 2537110 T^{16} + 8017759 T^{18} + 2537110 p^{2} T^{20} + 722131 p^{4} T^{22} + 183952 p^{6} T^{24} + 41485 p^{8} T^{26} + 8149 p^{10} T^{28} + 1357 p^{12} T^{30} + 61 p^{15} T^{32} + 2 p^{18} T^{34} + p^{18} T^{36} \) | |
| 7 | \( 1 + 41 T^{2} + 920 T^{4} + 14876 T^{6} + 194147 T^{8} + 2160581 T^{10} + 21160224 T^{12} + 186038921 T^{14} + 1487289488 T^{16} + 10876546309 T^{18} + 1487289488 p^{2} T^{20} + 186038921 p^{4} T^{22} + 21160224 p^{6} T^{24} + 2160581 p^{8} T^{26} + 194147 p^{10} T^{28} + 14876 p^{12} T^{30} + 920 p^{14} T^{32} + 41 p^{16} T^{34} + p^{18} T^{36} \) | |
| 11 | \( ( 1 + p T + 98 T^{2} + 633 T^{3} + 3702 T^{4} + 18273 T^{5} + 83643 T^{6} + 338749 T^{7} + 1288882 T^{8} + 4408692 T^{9} + 1288882 p T^{10} + 338749 p^{2} T^{11} + 83643 p^{3} T^{12} + 18273 p^{4} T^{13} + 3702 p^{5} T^{14} + 633 p^{6} T^{15} + 98 p^{7} T^{16} + p^{9} T^{17} + p^{9} T^{18} )^{2} \) | |
| 17 | \( 1 + 9 p T^{2} + 12140 T^{4} + 38925 p T^{6} + 27647744 T^{8} + 936619021 T^{10} + 26573077967 T^{12} + 643864711109 T^{14} + 13481718271300 T^{16} + 245509351760496 T^{18} + 13481718271300 p^{2} T^{20} + 643864711109 p^{4} T^{22} + 26573077967 p^{6} T^{24} + 936619021 p^{8} T^{26} + 27647744 p^{10} T^{28} + 38925 p^{13} T^{30} + 12140 p^{14} T^{32} + 9 p^{17} T^{34} + p^{18} T^{36} \) | |
| 19 | \( ( 1 + 14 T + 159 T^{2} + 1268 T^{3} + 9453 T^{4} + 3160 p T^{5} + 359828 T^{6} + 1898232 T^{7} + 9431345 T^{8} + 42172868 T^{9} + 9431345 p T^{10} + 1898232 p^{2} T^{11} + 359828 p^{3} T^{12} + 3160 p^{5} T^{13} + 9453 p^{5} T^{14} + 1268 p^{6} T^{15} + 159 p^{7} T^{16} + 14 p^{8} T^{17} + p^{9} T^{18} )^{2} \) | |
| 23 | \( 1 + 212 T^{2} + 22156 T^{4} + 1525094 T^{6} + 78118919 T^{8} + 3195297084 T^{10} + 109509883013 T^{12} + 3259103984113 T^{14} + 86493635472333 T^{16} + 2082372879567825 T^{18} + 86493635472333 p^{2} T^{20} + 3259103984113 p^{4} T^{22} + 109509883013 p^{6} T^{24} + 3195297084 p^{8} T^{26} + 78118919 p^{10} T^{28} + 1525094 p^{12} T^{30} + 22156 p^{14} T^{32} + 212 p^{16} T^{34} + p^{18} T^{36} \) | |
| 29 | \( ( 1 + 10 T + 191 T^{2} + 1715 T^{3} + 17911 T^{4} + 138653 T^{5} + 1066226 T^{6} + 6969195 T^{7} + 43714778 T^{8} + 240558293 T^{9} + 43714778 p T^{10} + 6969195 p^{2} T^{11} + 1066226 p^{3} T^{12} + 138653 p^{4} T^{13} + 17911 p^{5} T^{14} + 1715 p^{6} T^{15} + 191 p^{7} T^{16} + 10 p^{8} T^{17} + p^{9} T^{18} )^{2} \) | |
| 31 | \( ( 1 + 16 T + 338 T^{2} + 3654 T^{3} + 44527 T^{4} + 366176 T^{5} + 3254336 T^{6} + 692486 p T^{7} + 150914718 T^{8} + 815332320 T^{9} + 150914718 p T^{10} + 692486 p^{3} T^{11} + 3254336 p^{3} T^{12} + 366176 p^{4} T^{13} + 44527 p^{5} T^{14} + 3654 p^{6} T^{15} + 338 p^{7} T^{16} + 16 p^{8} T^{17} + p^{9} T^{18} )^{2} \) | |
| 37 | \( 1 + 351 T^{2} + 60259 T^{4} + 6727523 T^{6} + 549525954 T^{8} + 35185107606 T^{10} + 1858401521943 T^{12} + 84701180350315 T^{14} + 3473342101230435 T^{16} + 132423399397111738 T^{18} + 3473342101230435 p^{2} T^{20} + 84701180350315 p^{4} T^{22} + 1858401521943 p^{6} T^{24} + 35185107606 p^{8} T^{26} + 549525954 p^{10} T^{28} + 6727523 p^{12} T^{30} + 60259 p^{14} T^{32} + 351 p^{16} T^{34} + p^{18} T^{36} \) | |
| 41 | \( ( 1 + 26 T + 524 T^{2} + 7556 T^{3} + 94019 T^{4} + 983838 T^{5} + 9217055 T^{6} + 76031791 T^{7} + 569697537 T^{8} + 3819077279 T^{9} + 569697537 p T^{10} + 76031791 p^{2} T^{11} + 9217055 p^{3} T^{12} + 983838 p^{4} T^{13} + 94019 p^{5} T^{14} + 7556 p^{6} T^{15} + 524 p^{7} T^{16} + 26 p^{8} T^{17} + p^{9} T^{18} )^{2} \) | |
| 43 | \( 1 + 481 T^{2} + 112234 T^{4} + 17063848 T^{6} + 1914891279 T^{8} + 169986633537 T^{10} + 12450595427612 T^{12} + 772051482005341 T^{14} + 41142596153699380 T^{16} + 1898787455635830505 T^{18} + 41142596153699380 p^{2} T^{20} + 772051482005341 p^{4} T^{22} + 12450595427612 p^{6} T^{24} + 169986633537 p^{8} T^{26} + 1914891279 p^{10} T^{28} + 17063848 p^{12} T^{30} + 112234 p^{14} T^{32} + 481 p^{16} T^{34} + p^{18} T^{36} \) | |
| 47 | \( 1 + 483 T^{2} + 120667 T^{4} + 20376735 T^{6} + 2587235093 T^{8} + 261176578547 T^{10} + 21672451253682 T^{12} + 1509531591909833 T^{14} + 89408561635686157 T^{16} + 4535492562804574495 T^{18} + 89408561635686157 p^{2} T^{20} + 1509531591909833 p^{4} T^{22} + 21672451253682 p^{6} T^{24} + 261176578547 p^{8} T^{26} + 2587235093 p^{10} T^{28} + 20376735 p^{12} T^{30} + 120667 p^{14} T^{32} + 483 p^{16} T^{34} + p^{18} T^{36} \) | |
| 53 | \( 1 + 683 T^{2} + 227452 T^{4} + 49131109 T^{6} + 7727671456 T^{8} + 941993350203 T^{10} + 92466126532807 T^{12} + 7492788512562805 T^{14} + 509268798333100276 T^{16} + 29295973576532880736 T^{18} + 509268798333100276 p^{2} T^{20} + 7492788512562805 p^{4} T^{22} + 92466126532807 p^{6} T^{24} + 941993350203 p^{8} T^{26} + 7727671456 p^{10} T^{28} + 49131109 p^{12} T^{30} + 227452 p^{14} T^{32} + 683 p^{16} T^{34} + p^{18} T^{36} \) | |
| 59 | \( ( 1 + 38 T + 1044 T^{2} + 20558 T^{3} + 336268 T^{4} + 4581406 T^{5} + 54380633 T^{6} + 562232484 T^{7} + 5165090248 T^{8} + 41971149020 T^{9} + 5165090248 p T^{10} + 562232484 p^{2} T^{11} + 54380633 p^{3} T^{12} + 4581406 p^{4} T^{13} + 336268 p^{5} T^{14} + 20558 p^{6} T^{15} + 1044 p^{7} T^{16} + 38 p^{8} T^{17} + p^{9} T^{18} )^{2} \) | |
| 61 | \( ( 1 - 4 T + 343 T^{2} - 1723 T^{3} + 57309 T^{4} - 329369 T^{5} + 6181002 T^{6} - 37496025 T^{7} + 486972360 T^{8} - 2791463317 T^{9} + 486972360 p T^{10} - 37496025 p^{2} T^{11} + 6181002 p^{3} T^{12} - 329369 p^{4} T^{13} + 57309 p^{5} T^{14} - 1723 p^{6} T^{15} + 343 p^{7} T^{16} - 4 p^{8} T^{17} + p^{9} T^{18} )^{2} \) | |
| 67 | \( 1 + 637 T^{2} + 206705 T^{4} + 45083958 T^{6} + 7401121927 T^{8} + 973133933321 T^{10} + 106575454597536 T^{12} + 9977376091684953 T^{14} + 811953709927462434 T^{16} + 57984231073836644531 T^{18} + 811953709927462434 p^{2} T^{20} + 9977376091684953 p^{4} T^{22} + 106575454597536 p^{6} T^{24} + 973133933321 p^{8} T^{26} + 7401121927 p^{10} T^{28} + 45083958 p^{12} T^{30} + 206705 p^{14} T^{32} + 637 p^{16} T^{34} + p^{18} T^{36} \) | |
| 71 | \( ( 1 + 36 T + 996 T^{2} + 19836 T^{3} + 336537 T^{4} + 4788226 T^{5} + 60339266 T^{6} + 666679736 T^{7} + 6634195188 T^{8} + 58762523892 T^{9} + 6634195188 p T^{10} + 666679736 p^{2} T^{11} + 60339266 p^{3} T^{12} + 4788226 p^{4} T^{13} + 336537 p^{5} T^{14} + 19836 p^{6} T^{15} + 996 p^{7} T^{16} + 36 p^{8} T^{17} + p^{9} T^{18} )^{2} \) | |
| 73 | \( 1 + 703 T^{2} + 249479 T^{4} + 59638833 T^{6} + 10800612286 T^{8} + 1576053365150 T^{10} + 192035432281938 T^{12} + 19961668041313215 T^{14} + 1793109198399367608 T^{16} + \)\(14\!\cdots\!34\)\( T^{18} + 1793109198399367608 p^{2} T^{20} + 19961668041313215 p^{4} T^{22} + 192035432281938 p^{6} T^{24} + 1576053365150 p^{8} T^{26} + 10800612286 p^{10} T^{28} + 59638833 p^{12} T^{30} + 249479 p^{14} T^{32} + 703 p^{16} T^{34} + p^{18} T^{36} \) | |
| 79 | \( ( 1 - 8 T + 452 T^{2} - 3870 T^{3} + 108445 T^{4} - 872212 T^{5} + 16881900 T^{6} - 122568008 T^{7} + 1847644286 T^{8} - 11618283252 T^{9} + 1847644286 p T^{10} - 122568008 p^{2} T^{11} + 16881900 p^{3} T^{12} - 872212 p^{4} T^{13} + 108445 p^{5} T^{14} - 3870 p^{6} T^{15} + 452 p^{7} T^{16} - 8 p^{8} T^{17} + p^{9} T^{18} )^{2} \) | |
| 83 | \( 1 + 891 T^{2} + 398844 T^{4} + 118845506 T^{6} + 26401141787 T^{8} + 4645446406917 T^{10} + 671646033308264 T^{12} + 81682388971199473 T^{14} + 8479811819516364304 T^{16} + \)\(75\!\cdots\!37\)\( T^{18} + 8479811819516364304 p^{2} T^{20} + 81682388971199473 p^{4} T^{22} + 671646033308264 p^{6} T^{24} + 4645446406917 p^{8} T^{26} + 26401141787 p^{10} T^{28} + 118845506 p^{12} T^{30} + 398844 p^{14} T^{32} + 891 p^{16} T^{34} + p^{18} T^{36} \) | |
| 89 | \( ( 1 + 47 T + 1533 T^{2} + 34938 T^{3} + 653111 T^{4} + 10028303 T^{5} + 134744476 T^{6} + 1587555155 T^{7} + 17076113126 T^{8} + 167057679621 T^{9} + 17076113126 p T^{10} + 1587555155 p^{2} T^{11} + 134744476 p^{3} T^{12} + 10028303 p^{4} T^{13} + 653111 p^{5} T^{14} + 34938 p^{6} T^{15} + 1533 p^{7} T^{16} + 47 p^{8} T^{17} + p^{9} T^{18} )^{2} \) | |
| 97 | \( 1 + 997 T^{2} + 496684 T^{4} + 165132587 T^{6} + 41203780138 T^{8} + 8209449812665 T^{10} + 1354828636406625 T^{12} + 189477756993895903 T^{14} + 22774496209939633864 T^{16} + \)\(23\!\cdots\!08\)\( T^{18} + 22774496209939633864 p^{2} T^{20} + 189477756993895903 p^{4} T^{22} + 1354828636406625 p^{6} T^{24} + 8209449812665 p^{8} T^{26} + 41203780138 p^{10} T^{28} + 165132587 p^{12} T^{30} + 496684 p^{14} T^{32} + 997 p^{16} T^{34} + p^{18} T^{36} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{36} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.32648732790740781063133937429, −2.28451622830075972588763633028, −2.27626898838758561048592062466, −2.23302820847701560659712758390, −2.16074366095528945959299157140, −2.07911078634961001819027270844, −1.99031131306590998934389167328, −1.98241264647123904647264398865, −1.97435168188101915737659595400, −1.89159551076207999206713441501, −1.71119854907547789069675037046, −1.61799292200298596675874358169, −1.60408573581749634251150296169, −1.58290523472634107585048314234, −1.54603611874163039713044369232, −1.52040005389671110783951456353, −1.50495833783005447331856084351, −1.33674796641497315126502759761, −1.29876333122123602775032625165, −1.18911262590771112740047617483, −1.15706625835751581628955556729, −1.07803170981430337202910327274, −0.947868956406344699208139504372, −0.839864613521452601781121524197, −0.822127242833216050410273021544, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.822127242833216050410273021544, 0.839864613521452601781121524197, 0.947868956406344699208139504372, 1.07803170981430337202910327274, 1.15706625835751581628955556729, 1.18911262590771112740047617483, 1.29876333122123602775032625165, 1.33674796641497315126502759761, 1.50495833783005447331856084351, 1.52040005389671110783951456353, 1.54603611874163039713044369232, 1.58290523472634107585048314234, 1.60408573581749634251150296169, 1.61799292200298596675874358169, 1.71119854907547789069675037046, 1.89159551076207999206713441501, 1.97435168188101915737659595400, 1.98241264647123904647264398865, 1.99031131306590998934389167328, 2.07911078634961001819027270844, 2.16074366095528945959299157140, 2.23302820847701560659712758390, 2.27626898838758561048592062466, 2.28451622830075972588763633028, 2.32648732790740781063133937429
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.